In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of $mathbb{Z}_{m}*mathbb{Z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

A finite graph product is the fundamental group of a finite graph of groups. Finite graph products with finite cyclic edge groups are shown to inherit a solvable conjugacy problem from their vertex groups under certain conditions, of which the most important is that all the edge group generators in each vertex group are powers of a common central element.

LetG be a group. ThenG is cyclic if there exists some g ∈ G such thatG = 〈g〉 := {g : m ∈ Z}. For every positive integer n, Z/〈n〉 (the additive group of integers modulo n) is the unique cyclic group on n elements, and Z is the unique infinite cyclic group (up to isomorphism). The cyclic groups play a nontrivial role in abelian group theory. For instance, The Fundamental Theorem of Finitely Gener...

Analogous to the way the integers can be decomposed uniquely into prime factors, finite groups can be decomposed into finite simple groups, which cannot be further decomposed nontrivially. We will prove the JordanHölder Theorem and show that finite groups can always be decomposed uniquely into finite simple groups, and discuss the classification of these groups. In particular, we will discuss t...

We determine completely the structure of finite 2-groups which possess exactly six cyclic subgroups of order 4. This is an exceptional case because in a finite 2-group is the number of cyclic subgroups of a given order 2n (n ≥ 2 fixed) divisible by 4 in most cases and this solves a part of a problem stated by Berkovich. In addition, we show that if in a finite 2-group G all cyclic subgroups of ...

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than n/2 in the additive group Zn of integers modulo n. The main result states that for each zero-free sequence (ai) l i=1 of length l > n/2 in Zn ther...

Results of Lind on Lehmer’s problem include the value of the Lehmer constant of the finite cyclic group Z/nZ, for n ≤ 5 and all odd n. By complementary observations we determine the Lehmer constant of Z/nZ, for all n except for multiples of 420.

A permutation π of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a, b, c) is a non-trivial 3-term AP in G, that is c − b = b − a and a, b, c are not all equal, then (π(a), π(b), π(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Zn, for all n 6∈ {2, 3, 5, 7}. Here we prove, as a special case of a more gene...

A sequence in the additive group Zn of integers modulo n is called n-zero-free if it does not contain subsequences with length n and sum zero. The article characterizes the n-zero-free sequences in Zn of length greater than 3n/2−1. The structure of these sequences is completely determined, which generalizes a number of previously known facts. The characterization cannot be extended in the same ...

In this short note we describe the finite groups G having |G| − 1 cyclic subgroups. This leads to a nice characterization of the symmetric group S3. In subgroup lattice theory, it is a usual technique to associate to a finite group G some posets of subgroups of G (see, e.g., [4]). One such poset is the poset of cyclic subgroups of G, usually denoted by C(G). Notice that there are few papers on ...