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A simple graph is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be an arbitrary prime. Folkman proved [Regular line-symmetric graphs, J. Combin. Theory 3 (1967) 215–232] that there is no semisymmetric graph of order 2p or 2p2. In this paper an extension of his result in the case of cubic graphs of order 20p2 is given. We prove that there is no connected cubic semisymmetric graph of order 20p2 or, equivalently, that every connected cubic edge-transitive graph of order 20p2 is necessarily symmetric.

Let G be a group and w(G) = {o(g)|g ∈ G} be the set of element orders of G. Let k ∈ w(G) and sk = |{g ∈ G|o(g) = k}|. Let nse(G) = {s|k 2 ∈ w(G)}. In this paper, we prove that if G is a group and G2(5) is the Chevalley simple group of type G2 over GF(5) such that nse(G) = nse(G2(5)}, then G Entity List G2(5). [ABSTRACT FROM AUTHOR]

In this paper, we construct, using computations with Magma, a ternary code C from a primitive permutation representation of degree 15 of the group PSL2(9) by Key-Moori Method 1. The code C is an optimal code invariant under the group Sq. We consider the action of the automorphism group Sq on C and its dual. By taking the support of any codeword ω of weight l and orbiting it under S6, 1-(15,l,kι) designs are obtained, where kι = l|ωsθ |/15. For any codeword, the structure of the stabilizer in Sq is determined and primitivity of Sq on each design is examined. It is shown that the complement of one of these designs is actually a new design D with parameters 2-(15, 7, 36). Moreover, Aut(D) = S6. [ABSTRACT FROM AUTHOR]

A finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple groups. [ABSTRACT FROM AUTHOR]

Let U6n = be a group of order 6n. In this paper tetravalent normal edge-transitive Cayley graphs on U6n are considered. In this way several nonequivalent normal edge-transitive Cayley graphs on U6n are obtained whose automorphism groups are given exactly. [ABSTRACT FROM AUTHOR]

In this paper, we determine the simple groups G = AB, where B is isomorphic to L3(4) and A isomorphic to an alternating or a symmetric group on n ≥ 5, letters. [ABSTRACT FROM AUTHOR]

An element of a group G is called semi-rational if there is a natural number m such that each generator of 〈x〉 belongs to the conjugacy class of G containing x or xm. If all elements of G are semi-rational, then G is called a semi-rational group. In this paper, we study semi-rational Frobenius groups G and obtain results concerning effect of semi-rationality property on the kernel and complement of G. In particular, we show that ∣π(G)∣ ≤ 5 which answers Problem 2 in [D. Chillag and S. Dolfi, Semi-rational solvable groups, J. Group Theory 13(4) (2010) 535-548] for semi-rational Frobenius groups. [ABSTRACT FROM AUTHOR]

Normal edge-transitive Cayley graph Cay(G,S) where G is the generalized quater-nion group Q4n of order 4n or a certain group V8n of order 8n is investigated. It is shown that up to isomorphism there is only one tetravalent normal edge-transitive Cayley graph when G ≅ Q4n is the generalized quaternion group and its automorphism group is found. In the case of V8n we show that there is no normal edge-transitive Cayley graph on V8n. [ABSTRACT FROM AUTHOR]

The study of fuzzy subhypergroups of the complete hypergroups is an important area of research in fuzzy structure theory. Since this structure relies on a certain decomposition of a group G,in this paper we consider the cyclic groups of orders p and p2 and prove that they have the required decomposition. Here we use some aspects of group theory to solve a problem in the fuzzy algebraic hyperstructure theory. [ABSTRACT FROM AUTHOR]

Maturated and unmaturated groups were introduced by the Japanese chemist Shinsaku Fujita, who used them in the markaracter table and the Q-conjugacy character table of a finite group. He then applied his results in this area of research to enumerate isomers of molecules. Using the non-rigid group theory, it was shown by the second author that the full non-rigid (f-NRG) group of 2,3-dimethylbutane is isomorphic to the group (Z3×Z3×Z3×Z3):Z2 of order 162 with 54 conjugacy classes. Here (Z3×Z3×Z3×Z3):Z2 denotes the semi direct product of four copies of Z3 by Z2, where Zn is a cyclic group of order n. In this paper, it is shown with the GAP program that this group has 30 dominant classes (similarly, Q-conjugacy characters) and that 24 of them are unmatured (similarly, Q-conjugacy characters such that they are the sum of two irreducible characters). Then, the Q-conjugacy character table of the unmatured full non-rigid group 2,3-dimethylbutane is derived. [ABSTRACT FROM AUTHOR]

The maturated and unmaturated groups have been introduced by S. Fujita who used them in the markaracter table and the Q-conjugacy character table of a finite group. Fujita introduced more concise forms called the Q-conjugacy characters with integer-valued of the irreducible characters of finite groups and applied his results in this area of research to enumerate isomers of molecules. In this paper using GAP program all integer-valued characters of the full non-rigid group (f-NRG) of tetramethylethylene (2,3-dimethylbut-2-ene) is calculated by the Q-conjugacy relationships. It is shown that this group has 29 dominant classes (similarly, Q-conjugacy characters) such that 16 of them are unmaturated (similarly, Q-conjugacy characters such that they are the sum of two irreducible characters). Then the markaracter table and Q-conjugacy character table of the f-NRG of tetramethylethylene are derived for the first time. [ABSTRACT FROM AUTHOR]

Using nonrigid group theory, the full nonrigid (f-NRG) group of 1,3,5-trimethylbenzene (TMB) is shown to be isomorphic to the group S3[C3] = C3 S3 of order 162, where denotes the wreath product of groups, and C3 is the cyclic group of order three and S3 is the symmetric group of order six on three letters. This group has 22 conjugacy classes and irreducible representations. The character table of the full nonrigid TMB is then derived for the first time. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [ABSTRACT FROM AUTHOR]