Some finite solvable groups with no outer automorphisms

Gagen, T. M.

doi:10.1016/0021-8693(80)90239-2

Cited by 9 publications

(5 citation statements)

References 5 publications

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“…We deduce from (2) and our corollary to Lemma 3 that (5) A* is a p-group for some prime p and now Z(A) = 1 yields (6) F(A) is a p-group. …”

Section: Lemma 6 Suppose That a Satisfies The Following Conditions (mentioning

confidence: 61%

“…(2) A* is abelian, (3) {A*) w = (A*,t~lA*t) for some t in B, (4) A/F{A) is abelian, (see statements (7), (8), (9) and (10), pages 440-441 of [3]). We deduce from (2) and our corollary to Lemma 3 that (5) A* is a p-group for some prime p and now Z(A) = 1 yields (6) F(A) is a p-group.…”

Section: (I) Wr = (B*rmentioning

confidence: 61%

“…By symmetry, B/F(B) is cyclic as well. By induction hypothesis, (A*) w = (A*,t~1A*t) is the direct product of all iy-conjugate subgroups of A*, so, in particular, if x is an element of B, x 2 This contradiction shows that x 2 is the identity element, therefore we have Application of Lemma 6 shows that A is one of the excluded groups. So if A does not belong to this class of groups the induction step is successful and A* is normal in A G .…”

Section: It Follows That A/f(a)mentioning

confidence: 90%

“…We can now follow the proof of Theorem C in [3] in several steps, substituting our Lemma 3 for [3, Lemma 3], and we obtain (i) wr = (B*r, (2) A* is abelian, (3) {A*) w = (A*,t~lA*t) for some t in B, (4) A/F{A) is abelian, (see statements (7), (8), (9) and (10), pages 440-441 of [3]). We deduce from (2) and our corollary to Lemma 3 that (5) A* is a p-group for some prime p and now Z(A) = 1 yields (6) F(A) is a p-group.…”

Section: Lemma 6 Suppose That a Satisfies The Following Conditions (mentioning

confidence: 99%

“…We will first state and prove modifications of these lemmas, which in part are also more general when applied to the case of the class of all finite groups. The remainder (Lemmas 5 and 6) is needed to replace Gagen's Theorem [2]. LEMMA 1.…”

Section: Notationmentioning

confidence: 99%

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Subnormal embedding of relatively complete groups

Heineken

Soules

1988

J Aust Math Soc A

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In this paper we show that a group A is embedded in any finite group G as a subnormal subgroup with low degree of complication, provided that the automorphism group of A satisfies a condition depending on some Fitting class (which coincides with completeness for the Fitting class of all groups). A criterion is given for these groups as to whether they can be embedded subnormally in the commutator subgroup of some finite group or not.

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“…We deduce from (2) and our corollary to Lemma 3 that (5) A* is a p-group for some prime p and now Z(A) = 1 yields (6) F(A) is a p-group. …”

Section: Lemma 6 Suppose That a Satisfies The Following Conditions (mentioning

confidence: 61%

Section: (I) Wr = (B*rmentioning

confidence: 61%

Section: It Follows That A/f(a)mentioning

confidence: 90%

Section: Lemma 6 Suppose That a Satisfies The Following Conditions (mentioning

confidence: 99%

Section: Notationmentioning

confidence: 99%

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Subnormal embedding of relatively complete groups

Heineken

Soules

1988

J Aust Math Soc A

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On finite complete groups

Hartley

Robinson

1980

Arch. Math

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Finite groups

Кондратьев¹,

Махнев²,

Starostin³

1989

J Math Sci

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The present survey is formed primarily on the basis of works reviewed in the Referativnyi Zhurnal "Matematika" during 1976-1983 and is a continuation of surveys of the same name published in 1966, 1971, 1976 in the series "Albegra, Topologiya, Geometriya" (INT). Principal attention is devoted to finite simple groups and their classification.The present survey is formed primarily on the basis of works reviewed in RZh "Matematika" during 1976-1983 and is a continuation of surveys of the same title of A. I. Kostrikin [162], So A. Chunikhin, L. A. Shemetkov [371], and V. D. Mazurov [194]. Rapid development of the theory of finite groups was observed during this period. More than 2000 papers were published and also the books [

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Some finite solvable groups with no outer automorphisms

Cited by 9 publications

References 5 publications

Subnormal embedding of relatively complete groups

Subnormal embedding of relatively complete groups

On finite complete groups

Finite groups

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