Vol.30,1978 253

Abelian subgroups of large p-groups

By

goH~ D'ARcY*)

A theorem of Burnside [1 ; p. 226] states that if G is a group of order pn which has a center of order pc and a maximal normal abelian subgroup of order p m then

n < ~ i . i = c

Let us denote by N(p, m, c) the set of groups G which have the largest order compatible with having a center and a maximal normal abelian subgroup of orders

pc and pm respectively; that is ] G [ = pn where n = ~ i. This set is non-empty for i = c

all relevant p, m and c (p a prime and 1 --< c g m ) . :For example it is easily checked that the Sylow p-subgroup P of GL(m + 1, p) is in N(p , m, 1), while the procedure used in the proof of lemma 1 of this paper may be applied to P to find subgroups in •(p, m, c) for each c < m. When m > 2, the above examples have, in addition, abelian subgroups of orders greater than pro. The purpose of this paper is to show that in most cases this is a property of all the groups in N (p, m, c).

Theorem. Let G be in N (p, m, c) with c :4= m. Then i] m is greater than 3, G has an abelian subgroup o/order pm+ l.

Some observations:

1) When c ---- m, G is itself of order pm and thus has no abelian subgroup of order greater than pro.

2) The result does not hold for m • 2 (only the two non-abelian groups of order p3 are involved), or for m ---- 3 (see [2] p. 18, for all counterexamples in N(p, 3, 1)).

The theorem is proved through a series of lemmas.

Lemma 1. Let G be in N (p, m, c) where c < m. Then G contains a subgroup H that is in N ( p , m , m - 1).

P r o o f . We will in fact prove by induction that for each k between c and m, G has a subgroup in N (p, m, k). Assume G has a subgroup L such that L is in N (p, m, k -- 1).

*) This paper is part ofa Ph.D. thesis done at the University of,Notre Dame. I wish to thank Professor Karl M. Kronstein for his advice and interest.

254 J.D'ARcY ARCH. MATH.

Then letting r = ~ i , ILl--pe-~+~, [Z(L)I =p~-~ and L has a maximal normal i = k

abelian subgroup A of order pra. Let y be in A n Z2 (L) - - Z (L) and let H be the centralizer of y in L. Since y is in Z~ (L), the map x ~-~ [y, x] is a homomorphism from L into Z (L) with kernel H. Hence I L 1/I H I ~ [ Z (15) ] so tha t ]H I >= I L/Z (L) l =- =_ pr. On the other hand since clearly A C H and I Z(H)] >= I(Z(L), Y>I > p~, Burnside's theorem says that I HI --< Therefore I HI ---- and so again by Burn- side's theorem I z(H) 1 must be p~. Thus H is in N(p , m, k).

R e m a r k 1. Note tha t since IL/B I = I Z ( L ) I , the image of the map x ~> [y, x] is all of Z (L). Thus Z (L) is contained in the commutator subgroup K2 (L). We will need this fact later.

Lemma 2. Let G be a group o/nilpotency class >= 3 such that G/K3 (G) is extra- special. Then ] G/K3 (G) ] = pa.

P r o o f . The structure of G/K3(G) as an extra-special p-group implies tha t G has an even number of generators { x l , . . . , Xpr} such tha t [x~, xz] is in K3 (G) if {J, k} @- {2i - - 1, 2i} for some i = 1 . . . . . r while [x2~-1, xpd is in Ks(G) -- Ks(G). Let t be such tha t [xt, K~(G)] is not contained in K4(G). Then since for each i, [xpi-1, x2d generates K2 rood K3, [xei-x, x2~, xt] 4: e mod K4. Suppose r > 1. Then we can choose j so tha t t ~ 2 ] - - 1 or 2 j. By the Jaeobi identi ty

[x2j-1, xej, xt] [xp~, xt, x2i-1] [xt, x2~-1, x2i] -~ e rood K4.

The two commutators on the right are in K4 since we chose the generators so tha t [x2I, xt] and [xt, x2i-1] are in K3. Hence [xpj-1, xpj, xt] is also in K4 which contra- dicts our choice of t. Thus r ~ 1 and the result follows.

Lemma 3. I / G is in N(p , m, m -- 1), then either clG = 2 or m = 3.

P r o o f . Let A be the maximal normal abelian subgroup of G of order pm and let y generate A rood Z (G). Suppose clG = r > 2. Then Z2 (G) centralizes Kr-1. Since y is in Z2(G) and A is self-centralizing, we must have tha t A = Kr-1 = Z2(G). Now for any x in Kr-2 -- A, Ix, y] is non-trivial. Hence r - - 2 + r - - 1 _--< r, which implies tha t r =-3. Thus A = K2 and G/Z(G)= G/[A, G] = G/K3(G) is extra-special of order pm since IZ(G/Z(G))I = IA/Z(G) I -= p. So by Lemma 2 we have tha t m = 3.

R e m a r k 2. The group L generated by two elements, which is of exponent p > 3, of order p 5 and of class 3, belongs to N (p, 3, 2) and does not possess abelian subgroups of order p4. I ts central series structure is indicated in the proof of lemma 3.

Lemma 4. I / m is greater than 3, then every group in N(p , m, m - - 1) has an abelian (normal) subgroup of order pm+l.

P r o o f . We first show tha t every group G in ~r(p, m, m - - 1 ) , m4=3, is a so-called special p-group of class 2; tha t is Z (G) is elementary abelian and coincides with G', the commutator subgroup. G has class 2 by lemma 3, so G" C Z (G). Hence remark 1

Vol.30, 1978 Abelian subgroups of large p-groups 255

implies t h a t G' = Z (G). To show tha t Z (G) is e lementary abelian, let A and y be as in the previous lemma, and z be a ny element of Z(G). B y remark 1 there is an x in G such t h a t [y, x] = z. Therefore z~ = [y, x]~ = [y~, x] = e, since y~ is in Z(G).

Suppose no abelian subgroup had order pm+l. Then for each abelian subgroup B of G, BZ (G)/Z (G) would be either trivial or have order T- Now Heineken [3 *), Folge- rung to Satz 1] has shown t h a t i f a special T-group G has the proper ty tha t

.BZ (G)/Z (G) <= p

for all abelian subgroups B of G, then the rank of Z (G) is no t smaller t han the rank of G/Z(G). However by the definition of N(p, m, m -- 1), the rank of Z(G) is m - - 1, while G/Z (G) (which is also e lementary abelian since its exponent divides the expo- nent of Z(G) [4, Satz 2.13 a), p. 266]) has r ank m.

P r o o f o f t h e t h e o r e m . B y lemma 1, G contains a subgroup H which belongs to N ( p , m, m - - 1). L e m m a 4 says tha t H has a normal abelian subgroup of order pm+l when m is greater t han 3. Hence in t h a t case G contains an abelian subgroup of order 2m+1 as stated.

References

[1] W. BURNSIDE, On some properties of groups whose orders are powers of primes. Proe. London Math. Soe., II. Ser. 11, 225--245 (1912).

[2] J. D'AReY, p-groups with small abelian subgroups. Ph.D. dissertation, University of Notre Dame 1974.

[3] H. Hv~n~EKE~, Gruppen mit kleinen abelschen UntergTuppen. Arch. Math. 29, 20--31 (1977). [4] B. H~TPERT, Endliehe Gruppen I. Berlin 1967.

Anschrift des Autors:

John L. D'Arcy 3307 Oakmont Street Philadelphia, Pa. 19136, USA

Eingegangen am 24. 12. 1976

*) I am indebted to the referee for providing me with a preprint of [3].