Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

If p is a prime number, the poset of all nontrivial elementary abelian p -subgroups of a finite group plays an important role in both group theory and representation theory. It was studied by Quillen [9], who proved among many other things that it is homotopy equivalent to the poset of all nontrivial p -subgroups. In the case of a p -group P , one might believe that this poset has no interest s...

this paper, deals with some equivalence relations in fuzzy subgroups. further the probability of commuting two fuzzy subgroups of some finite abelian groups is defined.

In this note we give examples of a ring that has no maximal ideals. Recall that, by a Zorn’s lemma argument, a ring with identity has a maximal ideal. Therefore, we need to produce examples of rings without identity. To help motivate our examples, let S be a ring without identity. We may embed S in a ring R with identity so that S is an ideal of R. Notably, set R = Z⊕S, as groups, and where mul...

Let k[G] be the group algebra, where G is a finite abelian p-group and k is a field of characteristic p. A complete classification of finitely generated k[G]-modules is available only when G is cyclic, Cpn , or C2 × C2. Tackling the first interesting case, namely modules over k[C2 × C4], some structure theorems revealing the differences between elementary and non-elementary abelian group cases ...

We use the gluing construction introduced by Jia Huang to explore rings of invariants for a range modular representations. construct generating sets maximal parabolic subgroups finite symplectic group and their common Sylow p -subgroup. also investigate singular classical groups. introduce this compute invariant field fractions thin faithful representations semidirect products determine minimum...

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in X. If |X| ≥ |Y | for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset of pairwise non-commuting elements. In this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups GF of fields F containing a primitive pth root of unity. In this paper we present several constraints on such GF , using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these GF . In the case p > 2, each suc...