On groups in which every element has a prime power order and which satisfy some boundedness condition

We study the finite solvable groups 𝐺 in which every real element has prime power order. We divide our examination into two parts: the case O 2 ⁢ ( G ) > 1 \mathbf{O}_{2}(G)>1 and the case O 2 ⁢ ( G ) = 1 \mathbf{O}_{2}(G)=1 . Specifically we prove that if O 2 ⁢ ( G ) > 1 \mathbf{O}_{2}(G)>1 , then 𝐺 is a { 2 , p } \{2,p\} -group. Finally, by taking into consideration the examples presented in the analysis of the O 2 ⁢ ( G ) = 1 \mathbf{O}_{2}(G)=1 case, we deduce some interesting and unexpected results about the connectedness of the real prime graph Γ R ⁢ ( G ) \Gamma_{\mathbb{R}}(G) . In particular, we find that there are groups such that Γ R ⁢ ( G ) \Gamma_{\mathbb{R}}(G) has 3 or 4 connected components.

Nilpotency in automorphic loops of prime power order

Recently, graphs have started to be used to represent a finite ring. Nikmehr and Khojasteh in the article defined the nilpotent graph of a ring . Denoted , is a graph with the set of vertices being all the elements in the ring Two vertices and are adjacent if and only if is nilpotent elements in the ring . Topological index is a field that discusses graph structure based on the degree of each vertex of a graph and the distance between vertices. In this study, the author will gives the general formula of the Szeged index and Padmakar-Ivan index of the nilpotent graph graph of the modulo ring with prime power order. The result of this research is a general formula for the topological indices of nilpotent graphs of the integer modulo ring, called the Szeged index and the Padmakar-Ivan index.

We can now determine the structure of finite abelian groups. In particular, every such group is isomorphic to a direct product of cyclic groups, each having prime power order. The proof of this result is our main goal in the present chapter.

We show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.

Polynomial representations of complete sets of frequency hyperrectangles with prime power dimensions

Let G be a finite group. A gap G-module V is a finite dimensional real G-representation space satisfying the following two conditions: (1) The following strong gap condition holds: dimVP>2dimVH for all P<H≤G such that P is of prime power order, which is a sufficient condition to define a Gsurgery obstruction group and a G-surgery obstruction. (2)V has only one H-fixed point 0 for all large subgroups H, namely H∊L(G). A finite group G not of prime power order is called a gap group if there exists a gap Gmodule. We discuss the question when the direct product K×L is a gap group for two finite groups K and L. According to [5], if K and K×C2 are gap groups, so is K×L. In this paper, we prove that if K is a gap group, so is K×C2. Using [5], this allows us to show that if a finite group G has a quotient group which is a gap group, then G itself is a gap group. Also, we prove the converse: if K is not a gap group, then K×D2n is not a gap group. To show this we define a condition, called NGC, which is equivalent to the non-existence of gap modules.

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

The minimum genus problem consists on determining the minimum algebraic genus of a surface on which a given group G acts. For cyclic groups G this problem on bordered Klein surfaces was solved in 1989. The next step is to fix the number of boundary components of the surface and to obtain the minimum algebraic genus, and so the minimum topological genus. It was achieved for cyclic groups of prime and prime-power order in the nineties. In this work the corresponding results for cyclic groups of order N = 2q, where q is an odd prime, are obtained. There appear different results depending on the orientability of the surface. Finally, using the above mentioned results and those of this paper, we state explicitly, the general values for arbitrary number of boundary components, which are valid for each N < 12, and show how to deal with N = 12.

On groups with uncountably many subgroups of finite index

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.

We shall often need to consider the element Z =X Y, which is a parabolic transformation. Any parabolic transformation in r is conjugate to a power of Z. Let N be a normal subgroup of finite index ,u in r. The level n of N is defined as the least positive integer such that ZnCN. The conjugacy class of Z in r splits up into a finite number of equivalence classes under conjugacy by N. The number t of equivalence classes is called the parabolic class number of N. It is known that the integers ,u, n, t satisfy the relation

Generalizing results in [1], [3], [5], and [7] it is shown that the algebra generated by the quasi-regular representation λ of a discrete groupG onl2 (G/H) satisfies a standard polynomial identity if and only if λ (G) contains an abelian normal subgroup of finite index. An analogon for topological groups is also proved.

Let \(\mathcal{N}\) be a nonempty collection of normal subgroups of finite index of a group G and assume that \(\mathcal{N}\) is filtered from below, i.e., \(\mathcal{N}\) satisfies the following condition: $$\mbox{whenever }N_1,N_2\in \mathcal{N},\mbox{ there exists }N\in \mathcal{N}\mbox{ such that }N\le N_1\cap N_2.$$ Then one can make G into a topological group by considering \(\mathcal{N}\) as a fundamental system of neighborhoods of the identity element 1 of G. We refer to the corresponding topology on G as a profinite topology. If every quotient G/N\((N\in \mathcal{N})\) belongs to a certain class \(\mathcal{C}\), we say more specifically that the topology above is a pro - \(\mathcal{C}\)topology.