Normal subgroups contained in the Frattini subgroup
| Autor: | Charles R. B. Wright, W. Mack Hill |
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| Rok vydání: | 1972 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 35:413-415 |
| ISSN: | 1088-6826 0002-9939 |
| Popis: | If p is an odd prime and H is a p-group with a characteristic subgroup K such that |K| > IK n Z(H)J = p, then H cannot be a normal subgroup contained in the Frattini subgroup of any finite group G. We consider only finite groups. The order of the group G is IG!, Z(G) is the center of G, A(G) is the automorphism group of G and 1(G) is the group of inner automorphisms. If G is nilpotent, cl(G) denotes its nilpotence class. Other notation is also standard. Our aim is to prove the following Theorem. Let H be a p-group, p an odd prime, with a characteristic subgroup K such that IKI, 1> K nZ(H)I = p. Then H cannot be a normal subgroup contained in the Frattini subgroup of any finite group G. This result appears in [6] for arbitrary prime p, but under the additional hypothesis that cl(K) /2. It appears in [31 for the case that p is any prime and G is p-supersolvable. The case that IHI = 1|K = p3is covered in [5]. With no loss of generality (see [61), we take K = H and cl(JI) = 2. Then H is extra-special. For a discussion of extra-special p-groups and their automorphisms the reader is referred to [l], [7] and [8]. Our argument is based on two lemmas, the first of which is mentioned in [2]. (The author is grateful to Professor David Goldschmidt for a very helpful conversation concerning this result.) Lemma 1. If H is an extra-special p-group of exponent p, p odd, then A(H) splits over 1(H). Proof. H = (x1, x2, * ,X z) with x' =zP =. for each i and [xx21 = [x3, x4 = [x t x] z. Further, [xi, x.] 1 unless Ii, j} is one of l1, 21, 13, 4}, I.-, ln 1, n}. Each element of HI has unique representation as (Fln _xai)zb with 0 < a., b < p. If a e A(H), then for each i, a(x.) = (flnl x aii)zbi with (a..) e GL(n, p) and 0 < bi < p. Further, cg e I(H) if and only if (a..) is the identity matrix. Received by the editors December 16, 1974. AMS (MOS) subject classifications (1970). Primary 20D25; Secondary 20D15, 20D45. |
| Databáze: | OpenAIRE |
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