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M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where (among other things) the answer was shown to be affirmative if H has an Abelian Sylow 2-subgroup. Here, we show that the answer is affirmative unless the index of H is a power of a prime less than 5 and we obtain further restrictions on the structure of a purported minimal counterexample.
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It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.
We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove the main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases as well as torsion.
This paper is a new contribution to the study of regular subgroups of the affine group $AGL_n(F)$, for any field $F$. In particular we associate to any partition $lambda eq (1^{n+1})$ of $n+1$ abelian regular subgroups in such a way that different partitions define non-conjugate subgroups. Moreover, we classify the regular subgroups of certain natural types for $nleq 4$. Our classification is equivalent to the classification of split local algebras of dimension $n+1$ over $F$. Our methods, based on classical results of linear algebra, are computer free.
Let $G$ be a finite group and $sigma ={sigma_{i} | iin I}$ some partition of the set of all primes $Bbb{P}$, that is, $sigma ={sigma_{i} | iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_{i}$ and $sigma_{i}cap sigma_{j}= emptyset $ for all $i e j$. We say that $G$ is $sigma$-primary if $G$ is a $sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} leq A_{1} leq cdots leq A_{n}=G$ such that either $A_{i-1}trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $sigma$-primary for all $i=1, ldots, n$, modular in $G$ if the following conditions hold: (i) $langle X, A cap Z rangle=langle X, A rangle cap Z$ for all $X leq G, Z leq G$ such that $X leq Z$, and (ii) $langle A, Y cap Z rangle=langle A, Y rangle cap Z$ for all $Y leq G, Z leq G$ such that $A leq Z$. In this paper, a subgroup $A$ of $G$ is called $sigma$-quasinormal in $G$ if $L$ is modular and ${sigma}$-subnormal in $G$. We study $sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is $sigma$-primary.
We prove a general dichotomy theorem for groups of finite Morley rank with solvable local subgroups and of Prufer p-rank at least 2, leading either to some p-strong embedding, or to the Prufer p-rank being exactly 2.
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