We started our research with the intent on answering the following question: can we find a way to calculate all the subgroups of the symmetric group. This is easier said that done, as the number of subgroups for a symmetric group grows quickly with each successive symmetric group. This problem can actually be simplified to finding the subgroup conjugacy classes. So now we have the slightly diff...

Let α1,…,αr be algebraic numbers in a number field K generating subgroup of rank r K×. We investigate under GRH the primes p such that each orders (αimodp) lies given arithmetic progression associated to αi. also study for which index is fixed integer or set integers i. An additional condition on Frobenius conjugacy class may considered. Such results are generalizations theorem Ziegler from 200...

This paper defined the notion of ω , θ -complex fuzzy sets, id="M3"> subgroupoid, and id="M4"> subgroups described important examples under id="M5"> sets. Additionally, we discussed conjugacy class group with respect to id="M6"> norma...

We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that B...

3.3. The greatest common divisor and the least common multiple 12 3.4. Parabolic divisors 14 3.5. Divisibility: further properties 14 4. Normal forms arising from HNN extensions 16 4.1. HNN -normal form 16 5. Conjugacy problem 18 5.1. Conjugacy Criterion for HNN -extensions 18 5.2. Block decomposition 22 5.3. Conjugacy with respect to a subgroup 23 5.4. An algorithm for solving the Conjugacy Pr...

We prove that there exists a constant k with the property: if C is a conjugacy class of a finite group G such that every k elements of C generate a solvable subgroup then C generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4. We also present proofs that do not use the Classification theorem. The most direct proof gives a...

a finite group $g$ satisfies the on-prime power hypothesis for conjugacy class sizes if any two conjugacy class sizes $m$ and $n$ are either equal or have a common divisor a prime power. taeri conjectured that an insoluble group satisfying this condition is isomorphic to $s times a$ where $a$ is abelian and $s cong psl_2(q)$ for $q in {4,8}$. we confirm this conjecture.

For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2, R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKea...

We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup of shape 2.Co2. This is one of the first applications of a new general computational technique to enumerate big orbits. Mathematics Subje...

We survey recent computational results concerning the Monster sporadic simple group. The main results are: progress towards a complete classification of the maximal subgroups, including showing that L2(27) is not a subgroup; showing that the 196882-dimensional module over GF (2) supports a quadratic form; a complete set of explicit conjugacy class representatives; small representations of most ...