A well known theorem of G. A. Miller [4] (see also [2]) shows that a p-group of order p" where n > v(v 1)/2 contains an Abelian subgroup of order p° . It is clear that this theorem together with Sylow's Theorem implies that any finite group of large order contains an Abelian p-group of large order . In this note we use simple number theoretic considerations to make this implication more precise...

In this paper we generalize Kaplansky's combinatorial characterization of the isomorphism types embeddings a cyclic subgroup in finite abelian group given his 1951 book ``Infinite Abelian Groups''. For introduce partial maps on Littlewood-Richardson tableaux and show that they characterize direct sums such embeddings.

SO(3) it is the proper symmetry group of the icosahedron (and the dodecahedron). It is the unique finite subgroup of SO(3) which equals its own commutator subgroup—in fact, it is the unique non-abelian simple finite subgroup of SO(3). It can be thought of as the beginning or smallest member of the family of non-abelian finite simple groups. From the point of view of the McKay correspondence it ...

in this article, by using a partial on locally compact semi-direct product groups, we present a compatible extension of the fourier transform. as a consequence, we extend the fundamental theorems of abelian fourier transform to non-abelian case.

Let G be a group and let Autc(G) be the group of central automorphisms of G. We say that a subgroup H of a group G is c-characteristic if α(H) = H for all α ∈ Autc(G). We say that a group G is c-characteristically simple group if it has no non-trivial c-characteristic subgroup. If every subgroup of G is c-characteristic then G is called co-Dedekindian group. In this paper we characterize c-char...

We give efficient quantum algorithms for HIDDEN TRANSLATION and HIDDEN SUBGROUP in a large class of non-abelian groups including solvable groups of bounded exponent and of bounded derived series. Our algorithms are recursive. For the base case, we solve efficiently HIDDEN TRANSLATION in Z p , whenever p is a fixed prime. For the induction step, we introduce the problem ORBIT COSET generalizing ...

The Non-Abelian finite group PSL2(7) is the only simple subgroup of SU(3) with a complex three-dimensional irreducible representation. It has two maximal subgroups, S4 which, along with its own A4 subgroup, has been successfully applied in numerous models of flavor, as well as the 21 element Frobenius group Z7⋊Z3, which has gained much less attention. We show that it can also be used to generat...

We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, *, n)=(2(2&1) 3, 2(2+1) 3, 2(2+1) 3, 2) for d 0. The construction establishes that a McFarland difference set exists in ...

The critical group of an abelian network is a finite abelian group that governs the behavior of the network on large inputs. It generalizes the sandpile group of a graph. We show that the critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network. We exhibit the critical group as a quotient of a free abelian group by a subgroup containing th...

In this paper we study the structure of locally solvable, solvable, locally nilpotent, and nilpotent maximal subgroups of skew linear groups. In [S. Akbari et al., J. Algebra 217 (1999) 422–433] it has been conjectured that if D is a division ring and M a nilpotent maximal subgroup of D∗, then D is commutative. In connection with this conjecture we show that if F [M]\F contains an algebraic ele...