In this paper we study a representation of a fuzzy subgroup $mu$ of a group $G$, as a product of indecomposable fuzzy subgroups called the components of $mu$.  This representation is unique up to the number of components and their isomorphic copies. In the crisp group theory, this is a well-known Theorem attributed to Remak, Krull, and Schmidt. We consider the lattice of fuzzy subgroups and som...

We prove that a group G is Abelian whenever (1) it is nilpotent and the lattice of normal subgroups of G is isomorphic to the subgroup lattice of an Abelian group or (2) there exists a non-torsion Abelian group B such that the normal subgroup lattice of B × G is isomorphic to the subgroup lattice of an Abelian group. Using (2), it is proved that an Abelian group A can be determined in the class...

Exercise 1 (Maximal compact subgroups of G). A lattice in Qp is a finitelygenerated Zp-submodule of Qp that generates Qp as vector space. In particular, it’s free of rank n. Note that G acts transitively on the set of lattices in Qp . (i) Show that K = StabG(Zp ). (ii) Suppose that K ′ is a compact subgroup of G. Show that K ′ stabilises a lattice. (Hint: show that the K ′-orbit of Zp is finite...

In a 2001 paper, Shareshian conjectured that the subgroup lattice of a finite, solvable group has an EL-labeling. We construct such a labeling and verify that our labeling has the expected properties.

We prove that the subgroup lattices of finite alternating and symmetric groups do not contain so-called lower signalizer lattices in the class D. This result is one step in a program to show that the lattices in the class D are not isomorphic to an interval in the subgroup lattice of any finite group. In 1980 in [13], Palfy and Pudlak proved that the following two statements are equivalent: (...

Let π1(C) be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic p > 0 of countable cardinality. Let N be a normal (resp. characteristic) subgroup of π1(C). Under the hypothesis that the quotient π1(C)/N admits an infinitely generated Sylow p-subgroup, we prove that N is indeed isomorphic to a normal (resp. characteris...

In this paper, we have focused to study convex L-subgroups of an Lordered 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 about the set of all...