On p-groups with a cyclic commutator subgroup

J. Austral. Math. Soc. 20 (Series A) (1975), 178-198.

ON />-GROUPS WITH A CYCLIC COMMUTATOR SUBGROUP

R. J. MIECH

(Received 17 October 1972; revised 19 October 1973)

Communicated by G. Szekeres

This paper contains the complete classification of the finite p-groups G wherep is an odd prime, G is generated by two elements, and the commutator subgroupof G is cyclic.

These groups are a special kind of two-generator metabelian group, a classthat has been studied by Szekeres (1965). He determined the defining relations ofsuch groups but, as he noted, a "residual isomorphism problem" remains. Thecyclic commutator groups are simple when considered from this first point of view;they have a short, easily derived set of defining relations. However, the isomor-phism problem is a bit complicated for the defining relations contain nine par-ameters and each of these parameters might and can be an invariant of the group.

One particular type of cyclic commutator group, the metacyclic, has beenclassified. This was done by Basmaji (1969).

I am indebted to Professor Szekeres for his helpful comments, criticisms, andsuggestions on this paper.

The following notation will be used throughout this paper: (a, b,c, m, n, r,s)will denote nonnegative integers; capitals (R, S, M, N) will denote positive integersthat are relatively prime to p; p is an odd prime, p(r) = pr and p{xu •••,*„) = pm

where m is the minimum of xu---,xn.We need the defining relations of our groups to get under way. They are given

by:

THEOREM 1. Let p be an odd prime. Let G be a finite nonabelian p-groupgenerated by two elements and suppose that the commutator subgroup of G iscyclic. Then G has a pair of generators {x,y} such that the defining relations ofthe group are

[y,x] = z, xpW = zRp(r\ yp(b) = zSp(s)

zpM = 1, [z,x] = zMp(m\ [z,y] = zNpM

178

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[2] On p-groups 179

where a, b, c, r, s, m, n and R, S, M, N are integers that satisfy the conditions

a 2: b, a 2: c, r + m 2: c, r + n 2: c, s + n 2r c,

l ^ r a , n S c, 0 ^ r, s ^ c, p(b) - MSp(jn + s) = Omod^Cc).

Conversely given any set of parameters {a, b, c, r, s, m, n, R, S, M, N} that satisfiesthese conditions there is a group defined by these relations.

The defining relations stated in Theorem 1 are easy to derive: Let G2 be thecommutator subgroup of G and suppose that GjG2 = <G2x> ® <G2y> where<G2x> is of order p(a), <G2y> is of order p(b) and a 2; b. Then set z = [.y,x] andassume that G2 = <z> is of order p(c) to get the power relations. Finally once wesuppose that

[Z.JC] = zMp(m) and [z,y] = / " "

the higher order commutators are determined. We have, for example,

[Z,JC,X] = [zM^m),x] = zA

where A = (Mp(m))2. Note incidentally that a, b, and c are invariants of the group,G is of order p(a + b + c) and if q is the smallest integer such that

(q — l)min{m,n} 2: cthen G is of class q.

The inequalities and congruence given at the end of Theorem 1 will be derivedlater.

The notation arising in Theorem 1 can be abbreviated. The parameters a, b,and c will be considered fixed in what follows so the groups described there haveessentially four defining relations:

XPW = z « p « yPib) = zsP^ [ z > x - | = z*po»>f [ z > x - | = 2 * P «

We shall call this group "the group defined by [Rpr,Sps,Mpm,Npny\ Further-more in our classification scheme we shall be dealing with a number of groupswhere one of the parameters r,s,m, or n is equal to c; that is, a power or com-mutator is equal to 1. To make these cases more conspicuous the correspondingpc of the bracket notation will be replaced by 0. Thus the group defined by[0, Sps, pm, 0] is that group whose defining relations are

Finally, z will always denote the commutator of y and x; z = [_y,x].As for the isomorphism problem, our groups have four defining relations and

we get a system of four congruences that govern isomorphism between two suchgroups. The congruences are derived in section 1. Section 2 deals with the analysisof these congruences. The argument there splits into several cases for one has tomake assumptions about the relative sizes of r and 5 and m and n to carry out the



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180 R. J. Miech [3]

analysis. Once this is done one can write out the different isomorphism types; thereis quite a number of them.

The main factors that are employed in the presentation of the types in whatfollows are: (1) The size of b relative to c, i.e., b ^ c or b < c. (2) The relativesizes of a, b, and c, i.e., a - b > c, 1 ^ a — b ^ c, or a = b. (3) The number ofparameters among r, s, m, n, R, and S that appear as invariants in the defining rela-tions. The simplest case occurs when a, b, and c are spread apart. We then have:

THEOREM 2. If a — b > c and b ^ c the distinct (non-isomorphic) groups ofTheorem 1 having the parameters a,b, and c are given by.

(a) [Rpr,0,0, p"'] l^n^c, c-n^r^c, R ^ p(c - n, c - r).(b) [Rpr,0,pm,pn~] 1 ̂ m < n ^ c, c - m ^ r g c,

R 5S p(c — n, c — r).

(c) [Rpr,ps,0,pn] l ^ n ^ c , c - n g , s < r ^ c ,

R :g p{c — n, c — r).(d) [Rpr,Sps,pm,pn'] l g s ^ c - 1 , c - s g m g c - 1 ,

s + l ^ r ^ c , m + l ^ n r g c ,J? ^ p(c — n, c — r), S ^ p(r — s, n — m).

The following conventions will be used in the rest of this paper: max{qr, t} ^ uwill be abbreviated to {<?, f} sS u; similarly u ^ {<l,t} means that u ^ min{q, t}.In addition the "obvious" conditions, 1 ^ m, n ^ c and 0 ^ s, r ^ c will alwaysbe assumed but they will not always be mentioned.

THEOREM 3. Suppose that b ^ c and a — n = c — k where 0 ^ fc ^ c — 1.T/ien f/ie distinct groups associated with a,b, and c and which are defined interms of most four parameters are given by

(a) [pr,0,pm,0] 1 ̂ m ^ k, {c - m, c - k +m} <= r ^ c.(b) [0,ps,0,pn] kjl-^s-^k, c - s ^ n ^ c - k + s.(c) [R/,0,0,p"] 1 g n g c, c - n g r ^ c, R ^ p(c - n, c - r).(d) [0, Sps, pm, 0] c-k S s ^ k, c - s ^ m ^ k,

S 5S p(c — m, c — s).(e) [R/,0,pm,p"] l ^ r ^ c , c - r g j m ^ c - l ,

m + 1 ^ n g 2 c + m - f c - r - l ,i? ^ p(c — «, c — r, 2c + m — k — r — n).

(0 iRp^fAp"] l ^ n ^ c , 0 ^ s | c - l ,s + l ^ r ^ c - k - l + s , R £ p(c-r,c-n).

(g) [P r .S / ,p m ,0] 0 ^ j g c - l p s + l ^ r ^ c - 1 + s - fe ,c — r ^ m ^ k, S ^ j>(c — s, r + A: — m — s).

(h) [0,Sps,pm,pn] O ^ s ^ f e , c - s ^ m ^ c{c — s, s — fe + m + l } ^ n ^ c — fe+m — 1.S ^ p(n + k — s — m, c—tri)



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[4] On p-groups 181

THEOREM 4. Suppose that b ^ c and a — b = c — k where 1 ^ k ^ c — 2.Let st be the set of (r, s, m, ri) such that

l g s ^ c - 1 c — sf^m^c— I

s+l<Lr^c— 1 — k + s m + l^n^c— 1— k + m, r,n g c

Let $! be the subset of s/ where

s 2: k, m < k, and n > m + r — k.

Let @)2 be the subset of stf where

s < k, m~§.k, and n < k — s + r.

Let $)3 be the subset of s/ where

s < k and m < k.

Let 8$ denote the union of all the 39t. Then the distinct six parameter groups ofTheorem 1 associated with a, b, and c are given by [Rp,Sps,pm,p"'] where(r, s, m, n) is in s/,

R ^ p(c — r, c — r + s — k, c — n, c — n + m — k).and

S ^ p(r — s, n — m) for (r,s,m

S ^ p(r — s + k — m, n — m + k — s) for (r,s,m,ri)e<%.

Theorems 2, 3, and 4 deal with the case where b S; c and a — b 2; 1. The casea = b stands alone, which is not too surprising, for then there is a certain symmetrypresent in the denning relations. We have:

THEOREM 5. If a = b then the distinct groups of Theorem 1 havingparameters a, a, and c are given by

(a) [0,ps,0,pn] c/2 ^ s ^ c - 1, c-s i n ^ s.

(b) [0 ,S / ,p m ,0] O ^ s ^ c , c-s^m^c. S ^ p(c - m, c - s)

(c) [0, Sps,pm,pn~\ l ^ s g c - 1 , c-s ^ m ^ c{c — s, m + s — c + l } : g n : g m — 1

The next three theorems deal with the case b < c. These results could bencorporated into Theorems 2 through 4, but when one does this the inequalities3n the parameters become obscure.

THEOREM 6. Suppose b > c and a — b > c. Then the distinct groups of Theo-•em 1 associated with the parameters a, b, and c are given by [Rpr,Sps,pb~s,p"2vhere:



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182 R. J. Miech [5]

0 ^ s ^ ft — 1, c — b + s ^ r ^ c , c — s ^ n ^ c,

i? ^ p(c - n, c - r), S ^ p(r — s, b — s, n - b + s), S s l mod p{c — b).

THEOREM 7. Suppose that b < c and a — b = c — k where 0 5S k ^ b. Then

the distinct groups associated with a, b, and c which are defined in terms of at

most three parameters are given by:

(a) [O,Sps,pb-s,O~] b-k ^ s ^ {k,b - \), S = lmodp(c-b)

S ^ p{c - b + s, b - s).

(b) [ > r , S / , / - s , 0 ] b - k ^ s ^ b - l , c-b + s ^ r ^ c - k - l + s ,

S = 1 m o d p(c - b ) , S ^ p(b - s , r + k - b).

(c) [ 0 , S p * , p b - S , p n ~ ] b - k ^ s ^ k , { c - s , b - k + \ } ^ n - £ c - s+ b - k - l , S EE 1 modp(c - b)S ^ p(k + n - b, c + s - b, b - s).

THEOREM 8. Suppose that b < c and a - b = c — k where 2 ^ k ^ b. Lets# denote the set of(r,s,n) such that

0 g s g ft - 1, c-b + s ^ r ^ c - k - 1 + s

c — s ^ n ^ c — s + b — 1 — k, r,n ^ c.

Let 3SX be the subset of stf where

s ^ {k,b — k + 1}, r < b, and n>b-k + r-s.

Let 8%2 be the subset of s/ where

s ^ {fc - l,b - k} and n < {k - s + r,2b - 2s}.

Let S83 be the subset of sf where

b — k < s < k and r < b or n < 2b — 2s.

Let 3ft denote the union of all the 3S{. Then the distinct five parameter groups ofTheorem 1 associated with a,b, and c are given by [RpT,Sps,pb~s,p"'\ where(r, 5, n) is in 51, S = 1 mod p(c — b),

R :g p(c — r, c — r + s — k, c — n, c — n + b — s — k)and

S S P(r - s, n - b + s, b - s) for (r, s, n) 6 s/ - &,

S ^ p(r + k - b, n + k - b, b - s) for (r, s, ri) e 33.

This completes the tabulation of the groups of Theorem 1. Any such group isisomorphic to one of the groups of Theorems 2 through 8. Furthermore any twoclasses listed in any one of these theorems are disjoint.



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[6] On p-gvoups 183

1. The isomorphism congruences

Set

ix = 1 + Mpm, v = 1 + Np".

Note that /i is a function of M and m; v depends on N and n. Next, let

c t - l « - l

M(a) = 2 n', N(oc) = £ vf.i = 0 i = 0

We then have

THEOREM 9. Let G = <x, y} be defined by [Rpr, Sps, Mpm, Npn~\ andG = <jc, y} be defined by \_Rpr,5ps,M pm,Np"]. Suppose 9 is the mapping fromG to G defined by

xe = x'/z', f = xyydz".

Let A = a<5 — /fy. Then 9 is an isomorphism if and only if(A,p) = 1, y = y'pa~b

and

aRpT + pSps+a-b = s / j / m o d /

y'Rpr + dp" = eSps + ppb mod pc

/ iV = p.modpc

/iV s vmodpe

where

e = M

The number s appearing in Theorem 9 is prime to p: We have \i = v = 1 mod p,hence M(a) = a, N(^) = 8, •••, and e = A mod p.

We need several results on commutators to prove Theorem 9. The first is:

LEMMA 1. Let G be defined as in Theorem 1. Let \x and M(a) be defined as

above and set z(a) = z". Then

[/,*«] = z(M(a)N(/?))

PROOF. These results are easy consequences of the usual commutator relationsfor a metabelian group.

The inequalities stated in Theorem 1 can be derived at this point: By thedefining relations there,

1 = \_y*b\yl = [zSp(s),)>] = z s " p ( s + n ) .



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184 R. J. Miech [7]

Consequently, s + n 3; c. Next

1 = [z,/<"> ] = zA

where, by Lemma 1,

A = (1 + NpYw - 1 = Bpb+n

and (B,p) = 1. Thus b + n 7z c. Similar arguments with the power xp(a) yieldr + m 2: c and a + m ^ c.

To continue, by Lemma 1 and the inequality a + m ^ c, [y, xp(o)] = zA

where A = p" mod pc. But we also have

Consequently

(1) p(a) + NRp(n + r) = Omodp(c).

A similar congruence

(2) p(b) - NSpfjn + s) = Omodp(c)

can be obtained by expanding [x, yFm] in two ways. These two congruencesimply that a g: c and r + n ^ c: Suppose that a < c. Then by (1), a = n + r.Furthermore since b :g a < c we have by (2), b = m + s. Thus we have

a + b = (n + r) + (m + s) = (s + n) + (w + r) ^ c + c,

a contradiction. So, a ^ c and r + n S; c.Finally we must have m ^ 1 and n k 1. If for example n = 0 then [y,x]M

= I>,JC,X] so G2^G3 = [G,G2~].The converse to Theorem 1 can be proved in the conventional way.

LEMMA 2. Let G be defined as in Theorem 1. Then for any x,y in G andany positive integer q

(xyf = xyzGt t )

whereG(q) = S AIV.

This result can be proved by induction on q.Incidentally, it is known that if p is odd and G has a cyclic commutator sub-

group then G is regular Huppert ((1967; page 322)). Thus if q = p{b) in Lemma 2then G(p(b)) is a multiple of p(b).

The next result is about changing generators in the group.

LEMMA 3. Let G be defined as in Theorem 1. Suppose that

u = xVz*, v = xyysz".



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[8] On p-groups 185

Set

e = A#(a)JV(<5)/ - M(y)N(J})v'

X = (/v* - i)p - (pV - l)n

T = e + A.

Then, with z(a) = z",w = [>,«] = Z(T),

[w,«] = Z ( T [ / I V - 1]),

PROOF. This follows from Lemma 1.We are now in a position to prove Theorem 9. We adopt the notation of

Theorem 2 and suppose that 9, defined by

xB = u = x"/x", / = » = xry'z",

is an isomorphism from G to G in what follows.Going to the first defining relation of G and applying Lemma 3 we have

Then, since G is regular and p(a) ^ p(c), we have

= zA

where A = aRp(r) + pSp(s + a - b). Thus

(3) «Rp(r) + 0Sp(s + a - b) = xRp(f)modp{c)

Later on we shall show that this reduces to the first congruence of Theorem 2.Complications appear in the congruence from the second defining relation

since we might have b < c. Going one way

Going the other, and applying Lemma 2 several times along the route, we have

where

B = [M(y)iV((5) + pbiV

The last two sets of equations imply that xypW is in G2; consequently y = y'p(a — b).Bringing these results together we have

;4) y'Rp(r) + 5Sp(s) + B = i:5p(s)modp(c).

To continue we have



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186 R. J. Miech [9]

[y,x,x-]e = [«,«]*'<"> = z(TMp(m)),and

{y,x,x-\B = [v,u,u-] = z(T[yV-l]).

Since (x, p) = 1 it follows that

(5) //v" - 1 = Mp(/jj) mod p(c).

Similarly working with [ j , x, y] we get

(6) p V - 1 = Np(n) mod p(c).

Relations (5) and (6) are the last two congruences of Theorem 2; in additionthey can be used to simplify (3) and (4). Consider the right hand side of (3):

xp{f) = [£ + CuV - l)p - fciV - l)n]p(f).By (5) and (6),

CuV - l)p(f) = Mp(f + m) mod p(c)

(jxV - l)p{f) = Np(n + f)modp(c).

Then by the inequalities on the parameters for G, f + m ~§. c and r + n ^ c. Thus(3) reduces to the first congruence of Theorem 2.

To reduce (4) begin with

B = [M(y)N(S) + p(nyva - l)]G(p(b)).

Recall that since G is regular we have G(p(b)) = ip(b) for some integer i. Then by(6) and the fact that b + n ]> c we have

(p.yv* - l)G(p(b)) = iNp{b + n) = Omodp(c).

Furthermore since y = y'p(a — b) we have M(y)G(p(b)) = 0 mod p(c). ThusB = Omodp(c). As for the right hand side of (4):

tSp(s) = [e + OtV - l)p - Oft* - l)7t]Sp(s).

By (6) and the fact that n + s ^ c, (/*V - 1)K^) s 0 mod p(c). Then by (5) andthe congruence appearing in the defining conditions for G,

(jfvp - l)Sp(s) = MSp(m + s) = p(b)modp(b).

Thus

rSp(s) = s5p(s) + pp{b) mod p(c),

and (4) reduces to the second congruence of Theorem 2.This completes the proof of Theorem 9.Incidentally, the last two congruences of Theorem 2,



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[10] On p-groups 187

(1 + Mpmf(l + Np"f = (1 + Mpm) mod p(c)

(1 + Mpm)\l + Npn)s = (1 + Np") mod p(c),

are not difficult to analyze. For let K be the cyclic group of reduced residuesmodulo pc and Km be the subgroup of elements congruent to 1 modulo pm. Thenany integer of the form 1 + Mpm where (M, p) = 1 generates Km as the two con-gruences above form a simple system of equations in a cyclic group. This fact willbe used frequently, without further mention, in what follows.

We shall close this section with a list of three elementary transformations thatwill be useful in the sequel. But first we need

LEMMA 4. Let G be defined as in Theorem 1, e as in Lemma 3, andA = ad - Py. Set

= M £ h(MPmy-2.

v = 2 WThen

ep(r) = Ap(r) mod p(c)

ep(s) = Ap(s) + St(a)p(b) mod p(c).

To prove this write out the sums involved and use the conditions: r + n ^ c,r + m^.c,s + n^c, m + s = b if ft < c.

LEMMA 5. G = <x,j;> be defined by [Rpr,Sps,Mpm,Np"']. Then:

(a) If u = x* and v = y3 and (ad, p) — 1 then G = <u, v} and is defined by

[S'Rpir), a'Spis), n" - 1, v* - 1]

where d'd = lmodp(c — r), oca' = lmodp(c — s) if b 2: c and (a',p) = I ifb <c.

(b) If u = x and v = xyy where y = y'p(a — ft) then G = <w,u> and isdefined by

[Rp(r), y'Rp(f) + Sp(s), Mp(m), nyv - 1].

(c) If u = xyp and v = v then G = (u, v) and is defined by

[Rp(r) + pSp(s + a - ft), Sp(s), /tv" - 1, Np(nj] .

The results stated here are special cases of Theorem 2 for we can start with Gdefined by \Rpr, Sps, Mpm, Npn] and consider G as the group defined by the con-gruences. For example to obtain (a) set /? = y = n = p = 0 in Theorem 2. Thefirst congruence is then

aRp(r) = sRp(r) mod p(c).



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188 R. J. Miech [11]

Thus r = r and, by Lemma 4

ccRp(r) = a5Rp(r) mod

So, R = 5'R. Similarly, in the second congruence of Theorem 2 we have s = s and,by Lemma 4,

3Sp(s) = S[a8pis) + dt(x)p(by]modp(c).

Consequently, S = a'S where a' is the inverse of (a + t(<x)p(b — s)) modulop(c — s). Continuing this way, we get Lemma 5.

Note finally that if G is denned as in Theorem 1 then we may (and shall)assume that M = N = 1. This follows from Lemma 5 (a) for there are integersa and <5 with (ad, p) — 1 such that

(1 + Mpmf - 1 = pn, (1 + Nff - 1 s / m o d p e .

2. Point of the classification theorems

We shall assume that a > b throughout most of this sections; the case a = bwill be treated separately at the end. Note that if a < b then, in the notation ofTheorem 9, a and 8 are prime to p for y = y'pa~b, A = a<5 — fiy, and (A,p) = 1.

Next, we shall split the groups of Theorem 1 into three classes by means of thefollowing inequalities on r and s:

r ^ s, s<r<s + a — b, s + a — b^Lr^c.

Let !F be that class where r ^ s. Then b Si c, we may assume s = c, and r isan invariant of the group. For if b < c then, by the conditions of Theorem 1,b = m + s^.m + r'^,c. Consequently the second congruence of Theorem 9 is

pr(y'R + 5ps"0 = sSps mod pc,

and given d we can choose y' so that the left hand side here is 0 modulo p(c). Butthis means we can find an isomorphic image of G where s = c, i.e., we may assumes = c. Finally, by the first congruence of Theorem 9, r is an invariant of the group.

We shall first prove

THEOREM 10. Let J5" be those groups in Theorem 1 where r ^ s and a > b.Then b 2; c and the distinct elements of !F are given by:

(a) [R/ ,0 ,0 ,p n ] r + n^c, R ^ p(c - n, c - r).(b) \_Rpr,0,pm,pn'] r + m^c, m < n ^ q-l

q = c — r + m + a — b, R ^ p(c — r, c — n, q — n).(c) [pr,0,pm,0] r + m c, a^ -b ^ r - m.

The correspondence between this result and the theorems of the introduction



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[12] On /7-groups 189

is this: First assume that a — b > c. Then parts (a) and (b) of Theorem 10 yieldparts (a) and (b) of Theorem 2. When a — b = c — k where 0 ^ k ^ c — 1 thenlO.a corresponds to 3.c, lO.b to 3.e, and lO.c to 3.a.

PROOF OF THEOREM 10. As mentioned we can assume s = c and r is an in-variant of the group, so & is included in the set of groups G defined by\Rpr,0, pm, pn~\. Suppose G, defined by [Rpr,0, pm, pn] is isomorphic to G. ApplyingLemma 4 to the first two congruences of Theorem 9 we get

ocRp(r) = ARp(r) mod p(c), y'Rpif) s 0 mod p(c).

Since y = y'p(a — b) and, by the second congruence here, y' = y"p(c — r), wehave Ap(r) = (<x5 — Py)p(r) == cc3p(r) mod p(c). Consequently the isomorphismcongruences of Theorem 9 reduce to

R = SR mod p{c — r), y' = 0 mod p(c — r)

/**/ = p mod p(c), fiyvs = v mod p(c)

where n = 1 + pm, v = 1 + p", p. = 1 + pm, and v = 1 + pn.

We shall now split 2F into three par ts : Let #"x be the subset of J*" where

n ^ m, ^ 2 be the subset of J5" where m<n<m + a — b + c — r, and J5^ be

t h e s u b s e t w h e r e m + a — b + c — r^n-^c.

If n ^ m we may assume, by Lemma 5.c, that m = c in the defining relations.Thus J^i is included in the set of groups G defined by [Rpr,0,0, pn~]. Furthermoreif G, defined by [Rpr,0,0, pn], is isomorphic to G then, by the last congruence in(1), n = n. The isomorphism congruences for this case are then:

R = SRmodp(c - r), y' = 0modp{c — r)

vs = lmodp(c), v* s vmodp(c).

That last congruence of (2) implies that 3 = 1 mod p(c - n). Placing S — 1+ cop(c — ri) in the first congruence we get

R — R = coRp(c — n) mod p(c — r).

Thus if G and G are isomorphic then R = Rmodp(c — n, c — r). Conversely if Rand R are so related the congruences in (2) are solvable. In sum the distinctmembers of i*j are given by \Rpr, 0,0, p"] where r + n 3&c and R ^ p(c - n, c - r).

Let us go on to !F%, those groups where m<n<m + a — b + c — r. Since(CL5, p) = 1 and y = y" p(a — b + c - r) it follows, from (1), that m and n are in-variants for these groups. The isomorphism congruences are:

R = 5Rmodp(c — r), y' = 0modp(c - r)

Define f(m, n) modulo p(c — n) by



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190 R. J. Miech [13]

Then since y = y" p(a — b + c — r) the last congruence of (3) is equivalent to

8 — 1 = — y"f{m,ri)p(q — n)modp(c — n)

where q = m + a — b + c — r. Placing this in the first congruence of (3) we have

R - R = (-y"f(m,n)p(q - n) + cop(c ~ n))Rmodp(c - r).

Thus if the groups are isomorphic then R s R mod p(q — n, c — n, q — r). Sincethe converse also holds the distinct members of !F 2 are given by \Rpr, 0, pm, p"]where r, m, n, and R satisfy the conditions stated in Theorem 10.b.

The family ^3, those groups where m + a — b + c — r^Ln^c reduces tothe groups defined by [pr, 0, pm, 0]. To see this first apply Lemma 5.b to get n = c;then apply 5.a to get R = 1.

The classes J5",- are disjoint. Suppose first that ?F3 is not empty. That isa — b ^ r — m, a case that occurs when a — b g c — 1. Then the groups ofTheorem lO.b are distinct from those of lO.a or lO.c: For if n < c the members of10.b have generators that lie outside the centralizer of the commutator subgroup;the members of lO.a or lO.c have a generator that commutes with the commutatorsubgroup. If n = c the group from lO.b appears similar to the one from lO.c.However the condition c = n ^ q — 1 from lO.b implies that r — m < a — b;'mlO.c we have r — m ^ a — b. So, fF2

IS disjoint from ^r1 and J ^ . The last two

congruences of (1) show that J* x and J* 3 are disjoint, for suppose the group in# \ defined by \_Rpr, 0,0, p"~] is isomorphic to [ / , 0, pm, 0] from SFZ. Then, by (1)

(1 + ff = (1 + pm)modpc, (1 + pnf = lmod/.

Since (5, p) = 1 the last congruence implies that n = c. But then m = c. However,by the conditions given for J5^ in Theorem lO.c, m Sj r — (a — b). Since r ^ c anda — b ^ 1 we have a contradiction. Thus J ^ and J^3 are disjoint.

If J5^ is empty we have to show J ^ and #"2 are disjoint but this is an easyconsequence of (1) and the conditions given in Theorem 10.

This completes the proof of Theorem 10.

Let J^f denote the set of groups in Theorem 1 where s<r<s + a — b. Notethat the first two congruences of Theorem 9 imply that r and s are invariant forthese groups.

This case includes a particularly complicated subcase, the one that occurswhen m < n < m + a — b, so we shall introduce some notation for it. Let a — b= c - k. If b ^ c set

s# = {(r, s, m, n ) : s + m ^ c, 0 < r — s < c — k, 0 < n — m < c — k}.

If b < c set



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[14] On ^-groups 191

J / = {(r,s,m,n) :c — b S r — s < c — k, c — s ^ n < b + c — k — s, m = b — s}.

Next let

^ 1 = {(r> s,m,n) :m < k ^ s, r — n + m < k, r < b},

^2 = {(/> s,m,n):s < k ^ m, n — r + s < k, n — m < b — s},and

3SZ = {(r,s,m,n) :m, s < k, min{r — s, n — m} < b — s},

with the understanding that the last inequality for each of the J1; is to be sup-pressed if 6 Si c, and m = b — s if b < c. Let <M denote thelunion of all the 3d\.Note for later reference that each ^ , is empty when k ^ 0.

We can now state:

THEOREM 11. LetJf denote the set of groups in Theorem 1 where s < r < s+ a — b. Set m(n) = m — n + a — b and s(r) = s — r + a — b. Then the distinctelements of^C are given by:

(a) [Rpr,ps,0,pn] s + ntc, 0<r-s<a-b, b^cR rg p(c — r, c — n)

(b) [Rpr, Sps, pm, p"] (r, s, m,n)esZ, R^ p(c - r, c - n, m{n), s{r)),S :g p(r — s, n — m, b — s) on s/ — @)S 5S p(c — m — s(r), c — s — m(n), b — s) on &,Sp(m + s) = p(b) mod p(c)

(c) [pr,Sps,pm,0] r + m^c, 0<r-s<a-b, m^c-a + b,Sp(m + s) s p(b) mod p(c),S ^ p(c — s, c - m — s(r), b - s).

The correspondence between this result and the results stated earlier is this:If fc Si c and a — b > c then Theorem 11.a corresponds to Theorem 2.6, l l .b to2.d, and ll .c is empty. If b S: cand a — b = c — A: where 0 ^ k ^ c — 1 then 11.acorresponds to 3.f, l l .b to Theorem 4, and l l .c to 3g. If b < c and a — b > c thenll .b corresponds to Theorem 6. If b < c and a — b = c — k where 0 ^ k ^ c — 1the l l .b corresponds to Theorem 8 and l l .c to 7.b.

The proof of Theorem 11 is based on:

LEMMA 6. Suppose that 0<r-s<a — b, G is defined by [Rpr, Sps, pm, p"1],G is defined by [Rpr, Sps, pm, p"~}, and G is isomorphic to G. Then the congruencesof Theorem 9 are equivalent to the system:

R = SR - pSp(s + a - b - r)modp(c - r)

S = -y'Rp{r - s) + oiS + (t' + p)p(b - s)modp(c - s)

/ / / = p. mod p(c), nyvs = vmodp(c),

vhere t' is an integer that depends on S and a.



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192 R. J. Miech [15]

PROOF. First replace p in Theorem 9 by dp. Then, applying Lemma 4 to thefirst two congruences of Theorem 9 we get

aRp(r) + pSp(s + a - b) = ARp(r) mod p(c)

( 4 ) = ASp(s) + 5{t' + p)p{b) mod p(c),

when t' = t(a)5. Eliminating Rp(r) from the system one gets the congruence forS stated above. Then, placing this value of S in the first congruence of (4), we getthe congruence for R. Conversely if R and S are given by Lemma 6 then the rela-tions in (4) hold.

We begin the proof of Theorem 11 by considering 3^u that subset oDf wheren ;§ m. Note that since c ^ n + s £ j r o + swe must have b ^ c for this subcase.Suppose then that G is in Jfx and is defined by [Kpr, Sps, pm, p"]. Since n ^ m wemay assume, by Lemma 5.c, that m — c. Next, when b > c the number a' is theinverse of a so we can take S = 1. In short, the defining relations of a group in^f t

may be assumed to be of the form [Rpp, ps, 0, p"]. By Lemma 6 the distinct membersof the family are obtained (for fixed r, s, and n) by letting R run through thereduced residues from 1 to p{c — n, c — r).

The above argument yields the groups of part (a) of Theorem 11. We shallskip part (b) for the moment and go on to the last part. Let^f 3 be the subset of Jfwhere m + a — b^n^c starting with the usual defining relations we can assume,by Lemma 5.b and then Lemma 5.a that n = c and R = 1. Thus G in Jt?3 hasdefining relations of the form [pr,Sps,pm,0]. If G defined by [ / , Sps, pm, 0] isisomorphic to G then, by Lemma 6, m = m and

1 = (5 - /JSp(s + a - b - r)modp{c - r)

S = a.S - y'p(r - s) + ((' + p)p{b - s)modp(c - s)

p." = H mod p(c), n7 = 1 mod p(c).

The third and fourth congruence here are equivalent to

a = 1 + xp{c — m), y' = y" p(c - (a - b + m)).

Thus the second congruence is equivalent to

S - S = -cSp(c - m) - y"p{c - m - s(r)) + ((' + p)p(b - s)modp(c - s)

This shows that if G and G are isomorphic then, necessarily,

S s ,5modp(c — m — s(r), c — s, b — s).

Conversely if S and S are so related these congruences are solvable and the cor-responding groups are isomorphic.

The families Jf \ and Jf3 are disjoint: First if J*F3 is not empty then thedefining inequalty m + a — b ^ n g c is not vacuous; thus m < c for a > b.



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[16] On ^-groups 193

Second if we assume the group from ^ corresponding to [Rpr, ps, 0, p"] issomorphic to the one from Jf3 defined by [pr, Sps, pm, 0] then, by the last two

congruences of Lemma 6, m = c.We now turn to Theorem l l .b. This part corresponds to the class ^ 2 > those

groups where 0 < r — s < a — b and 0 < n — m < a — b. By Lemma 6, m andn are invariant for these groups. Thus if the exponential parameters (r, s, m, and ri)are fixed we have two other parameters, R and S, to consider. Generally speaking itis easy to find necessary conditions on R and S for isomorphism; complicationsarise in the sufficiency question.

We shall need:

LEMMA 7. Suppose that 0 < n — m < a — b. Let h(m,n) and f(m,n) bedefined modulo p(c — ri) by

^(m.nMnm) = v m o d

Hp("-m) = v/(m-n) mod p(c),

where n = 1 + pm and v = 1 + p". Let H = h(m, n) and

F = hin, m + a — b)f(m, m + a — b).

Then the last two congruences of Lemma 6 are equivalent to the system

a = 1 — fiHp(n — m) mod p(c — m),

5 s 1 — y'Fp(m + a — b — n)mod p(c — n).

Furthermore ifm + a — b^c then 1 — HF = mod p(c — m — a + b).

This result is easy to verify, so we omit the proof.Suppose then G is defined by R and S, G is defined by R S, and G is isomorphic

to G. Employing Lemmas 6 and 7 and recalling that m{ri) = m + a — b — n,s(r) = s + a — b — rwe have

R - R = -y'RFp{m{ri)) + cop(c - ri) - j8Sp(s{r))mod p(c - r),

S - S = -y'Rp(r - s) + tp(c - m) - pHSp(n - m)

+ ( ' ' + P)P(b — s) mod p(c — s),

where co and x are integers. Consequently if G is isomorphic to G thenR = Rmodp{f) and S = 5modp{g) where

/ = min{m(n), c - n, s(r), c - r},

g = min{r — s, n — m, b — s}.

However if R, R and S, S are so related the corresponding groups need not beisomorphic.



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194 R. J. Miech [17]

To go on to the converse problem suppose that R — R = y(\)p{f) andS — S = y(2)p(g) where XI) and y(2) are integers and/and g are denned as above.Then the isomorphism congruences for the groups are

= -y'RFp{m(n) + cop(c - n) - pSp(s(r))modp(c - r)

= -7'Mr - s) + ip(c - m) - /*tf,$p(n - m)

+ if' + P)p(b — s)modp(c - s).

Now if the groups under consideration were isomorphic then the system (5) wouldbe solvable for arbitrary values of XI) and Xs) . but this does not happen all thetime.

We proceed by cases.

CASE 1. s + a — b }z c and m + a — b — n^c — r: Under these circum-stances the first congruence of (5) reduces to

y(l)p(c — n, c — r) = cop(c — ri)modp(c — r),

which is solvable for any given value of XI)- The second congruence of (5) is alsosolvable for any given value of y{2). Suppose for example that

g = min{n — m, b — s, r — s} = r — s.

Go to the second congruence, choose and fix any % and fi, and then choose y' sothat

y(2)p{r — s) = —y'Rp(r — s) + xp(c — m) — ptiSp{n — m)modp(c — s).

Finally choose p so that

it' + p)p(b - s) = Omodp(c - s).

The argument for the other two cases, g = n — m or g = b — s, is similar. Thusthe conditions R = Rmodp(f) and S = 5modp(g) are sufficient here.

CASE 2. s + a — b ^ c and m + a — b — n < c — r: If the parameters satisfy

these inequalities the first congruence of (5) reduces to

y(l)p(m(n), c — n) = —y'RFp(m(n)) + a>p(c — n)modp(c — r),

which is solvable for any given XI)- However the variable one uses to solve it, y' orto, depends on the size of m + a — b relative to c. If m + a — b ^ c the con-gruences reduce to the forms that appeared on case 1 and, again, the conditionsR = R mod p(f) and S = S'mod p(g) are sufficient.

So suppose that m + a — b < c. Then the congruences of (5) take the form



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[18] On p-groups 195

y(l) = -y'RF + cop(c - (m + a - b)modp(c - r - m(n))

( 6 ) y(2)p(g) = -y'Rp(r - s) + xp{c - m) - pHSp(n - m)

+ (f + P)p(b - s) mod p(c - s).

Now the inequalities s + a — b ^ c and m + a — b — n < c — r imply that r — s< n — m so g = min{r — s, b — s}. If g = b — s the system (6) is solvable and theconditions on R, R and S, S are sufficient.

The final subcase of case 2 occurs when m + a — b < c and r < b. Accordingto the first congruence of (6) we can take y(l) = 0. Doing so and then eliminatingy' from (6) we get

y(2)F = zFp(c - m - r + s)- PFH5p(n - m - r + s) + (f' + p)Fp(b - r)

— cop(c — m — a + b) + Xp(c — r — m(n)) mod p(c — r).

That is y{2) = 0 mod p(h) where

h = min{n — m — r + s, b — r, c — m — a + b, c — r — m(n)}.

Recall that at this stage G is defined by R and S, G is defined by R andS, R = Rmodp(m(n)) and S = S + y(2)p(r - s). Thus if 0 ^ y(2) < p(h) then Gand G are not isomorphic; if y(2) 2i p(h) they are. Next if 0 g y(2) < p(h) then

l^S+ y(2)p(r - s) < p(r - s) + (p(h) - \)p(r - s) = p(h + r - s)where

h + r — s = min{n — m, b — s, c — m — s(r), c — s — m(n)}.

Finally we have

s + a — b 'S: c so n — m ^ c — s — (m + a — b — n) = c — s — m(n)

and the distinct groups of this subcase are given by the R and S where R g p(m(n))and S ^ p(b — s, c — s — m(n), c — m — s(r)).

Cases 1 and 2 deal with the congruences in (5) when s + a — b ^ c. The oneexception to the rule S ^ p{r — s, n — m, b — s) occurs in the last part of case 2when s + a — b^ c, m + a — b — n < c — r, m + a — b < c and r < b. Recallingthat a — b = c — k, the exceptional case occurs when

s > k, m — n + r < k, m < k and r < b .

This is the set J ^ that was defined in the paragraph preceeding the statement ofTheorem 11.

CASE 3. s + a— b < c and m + a — b }z c: Consider the first congruence in(5). Since m(n) = m + a — b — n ^ c — n the y' term there can be combined withthe co term to give a congruence of the form



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196 R. J. Miech [19]

y{l)p(c - n, s(r)) = cop(c - n) - p5p(s(r)) mod p(c - r).

Now if c — n ^ s(r) then the system consisting of this congruence and the secondcongruence of (5) is solvable and the necessary conditions are also sufficient.

So suppose that c — n 2; s(r). Then the congruences of (5) take the form

y(l) = cop(c — n — s(rj) — 05mod p{c — s — a + b)

y{2)p{g) = -y'Rp{r - s) + rp(c - m) - pHSp(n - m)

- s)modp(c - s).

Now the conditions m + a — b ^ c and c — n > s + a — b — r imply that n — m< r — s. Thus g = min{n — m, b — s}. If g = b — s this system is solvable andthe necessary conditions are sufficient. The final subcase of case 3 occurs whenn — m < b — s. The argument here is similar to the one given for the last part ofcase 2.

The total set of inequalities governing the exceptional subcase of case 3 is:

s + a — b < c, m + a — b ^ c, c — n }i s + a — b — r, n — m < b — s.

Setting a — b = c — k this translates to the set &2 that was defined earlier.

CASE 4. s + a — b ^ c and m + a — b < c: It is obvious that the system (5)is solvable if g = min{n — m, r — s, b — s] = b — s. So let us assume thatg ^ b —Is, say g = r — s. Then r — sf^n — m, m(n) ^ s(r) and (5) is equivalent to

s —y'RF + cop(c — m — a + b) — pSp{s — r + n — m)modp(c — r — m(n))

y(2) = -y'R + xp(c — m — r + s) — fiH5p(s — r + n - m)

+ (f + p)p(b - r)modp(c - r).

According to the first congruence we can assume that y(l) = 0. Doing so and theneliminating y' we get

y(2)F = tp(c -m-r + s)- cop(c - m - a + b) + pS(l - HF)p(n - m + s - r)

+ F(t' + p)p(b — r) + kp(c — r — m(n)) mod p(c — r).

Examining the exponents of p here we have c — m — r + s > c — m — a + b forwe have r — s < a — b. Furthermore, by the last part of Lemma 7, 1 — HF= 0modp(c — m — a + b). Thus y{2) = mod p(i) where

i = min{c — m — a + b, c — r — m{n), b — r}.

But then, as before, S = S + y(2)p(r — s), the underlying groups are notisomorphic if y(2) < p', and the distinct groups are obtained by letting S runthrough the reduced residues less than p(b — s, c — s — m{n), c — m — s(r)).



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[20] On /j-groups 1 9?

The argument for the subcase min{« - m, r - s, b - s} = n - m is similar to

tine one abo\e.FitvaWv, the dass 2? 2 ^ d-isjovat from $ ^ and 3»? 3.Th\s MYcm from \tae \ast

two congruence of Lemma 6 and the conditions on the parameters.This completes the proof of Theorem 11.

THEOREM 12. Suppose a > b and let 3? be the groups in Theorem 1 wheres + a~b^r^c. Then the distinct elements of ^ are given by:

(a) [0, ps, 0, />"] s + n^c, n^s + a-b, b^ c.(b) [0,Sps,pm,pn~] s + n^c, s + a-b-c<n — m<a-b

Sp(m + s) s p(b)modp(c)S ^ p(c — m, b — s, c — s — m — a + b + n)

(c) [0, Sps, pm, 0] m ^ c-a + b, Sp(m + s) = p(b) mod p(c)S ^ p(c — m, c — s, b — s).

The correspondence here to earlier theorems is this: If b Si c then 12.a cor"responds to 3.b, 12.b to 3.h and 12.c to 3.d. If b < c then 12.b corresponds to 7.cand 12.c to 7.a.

To start the proof of Theorem 12 we apply Lemma 5.c and conclude that 3? isincluded in the set of groups defined by [0,Sps,pm,p"'j. Furthermore, by Theorem9 and Lemma 4, the isomorphism congruences are

fiSp(s + a - b) = 0 mod p(c),

(7) 6Sp(s) = oc3Sp(s) + (St(a) + p)p(b) mod p{c\

fi"vfi = p. mod p(c), fiyva s vmodp(c).

Let £'l be those groups in 3? where c - (s + a — V) + n ^ m. Then, byLemma 5.c, we may assume that the defining relations for this subclass are[0, Sps, 0, p"]. Next we must have b ^ c for if b < c then m + s - b; but m = c.Finally since b ^ c the number a' of Lemma 5.a is the inverse of a modulo p, sowe may assume that a = 1.

Let y2 De those groups where m<c-(s + a-b) + n and n < m + a - b;

i.e., a-b-(c-s)<n-m<a-b. The congruences of (7) imply that m andn are invariants and the isomorphism congruences are

0 = p'p(c - s - a + b)

d(S - aS) = (St(x) + p)p(b - s)modp(c - s)

pi*'1 = v~pmodp(c), Vs'1 = n~ymodp(c).

Consequently,

a — 1 = —f]'Hp(c — s — a + b + n — m)modp(c — m)



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198 R. J. Miech [21]

where H is a number prime to p. So if the groups are isomorphic thenS s Smod p(i) where

i — min{c — s — a + b + n — m, c — m, b — s}

the converse also holds.Let JS?3 be those groups where a — b ^ n — m fL c. Then, by Lemma 5.b,

this class is included in the groups defined by [0, Sps, pm, 0] and it is not difficultto verify that the distinct groups correspond to the S where

S ^ p(c — m, c — s, b — s).

The usual arguments show that the classes J?; are disjoint.Let J f denote the set of groups of Theorem 1 when a — b. This case is similar

to the one that appeared in Theorem 12; some details of the proof are differentbut the final results are nearly identical. To begin J f is included in the set of groupsdefined by [0,Sps,pm,pn~\. To prove this start with G defined by [Rp',Sps,pm,pn'],apply Lemma 5.c to show that one may assume r 'Si s, and then apply it once moreto eliminate the p' term. Furthermore the isomorphism m congruences are thesame as those in (7). However the argument branches into two cases: s < c ands = c.

If s < c one gets three cases which are analogous to the JSf{ of Theorem 12.Furthermore, as long as one adds the condition s < c, the final results are the sameas those stated in Theorem 12. The cases of Theorem 12 yield those of Theorem5 for s < c.

If s = c one gets a set of groups defined by [0,0, pm, p"]. The subset here wheren ^ m can be reduced, by Lemma 5, to the [0,0, pm, 0 ] ; the subset where m < nto [0,0,0, p"]. Then one can show, by switching generators, that the groups of thelast two classes are isomorphic when m = n. In short the a = b, s = c casereduces to the set of groups [0,0, pm, 0] where 1 ^ m ^ c. This can be incor-porated into our final results by taking s = c in Theorem 5.b.

References

B. G. Basmaji (1969), 'On the isomorphisms of two metacyclic groups', Proc. Amer. Math. Soc.22, 175-182.

B. Huppert (1967), Endliche Gruppen (Die Grundlehren der math. Wissenschaften, Band 134Springer-Verlag, Berlin and New York, 1967).

G. Szekeres (1965), Metabelian groups with two generators, Proc. Inter. Conf. Theory of Groups,Austral. Nat. Univ., Canberra, 1965, 323-346; (Gordon and Breach, New York,1967).

Department of MathematicsUniversity of CaliforniaLos Angeles, 90024U. S. A.



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