We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x ∈ G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. To Charles Leedham-Green on his 65th birthday

We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence latt...

We present a polynomial time exact quantum algorithm for the hidden subgroup problem in $\Z_{m^k}^n$. The uses Fourier transform modulo $m$ and does not require factorization of $m$. For smooth $m$, i.e., when prime factors are size $(\log m)^{O(1)}$, can be exactly computed using method discovered independently by Cleve Coppersmith, while general Mosca Zalka is available. Even $m=3$ $k=1$ our ...

The main result of this paper states that for any group G with an automatic structure L with unique representatives one can construct a uniform partial algorithm which detects L-rational subgroups and gives their preimages in L. This provides a practical, not just theoretical, procedure for solving the occurrence problem for such subgroups. 1. Generalized word problem and rational structures on...

Szasz [l ] has recently shown that a group is cyclic if and only if it satisfies condition (A) below. (A) Every cyclic subgroup of the group is for some positive integer k the subgroup generated by the £th powers of the elements of the group. We shall extend this idea here to show that a metacyclic group whose commutator subgroup has order relatively prime to its index is characterized as a sol...

In this paper, we have focused to study convex $L$-subgroups of an $L$-ordered group. First, we introduce the concept of a convex $L$-subgroup and a convex $L$-lattice subgroup of an $L$-ordered group and give some examples. Then we find some properties and use them to construct convex $L$-subgroup generated by a subset $S$ of an $L$-ordered group $G$ . Also, we generalize a well known result a...

We prove that a finite group with nilpotent subgroup H and H-connected transversals is solvable. The proof depends on the classification of finite simple groups.

We prove a result on the asymptotic proportion of randomly chosen pairs $$(\sigma ,\tau )$$ permutations in symmetric group $$S_n$$ which “invariably” generate nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur same solvable subgroup . As an application, we obtain that for large degree “random” integer polynomial f, reduction modulo two different primes can be expecte...

We investigate subgroups of the group PLo(I) of piecewise-linear, orientation preserving homeomorphisms of the unit interval with finitely many breaks in slope, and also subgroups of Thompson’s group F . We find geometric criteria determining the derived length of any such group, and use this criteria to classify the solvable and non-solvable subgroups of PLo(I) and of F . Let H be a subgroup o...