OD-Characterization of alternating and symmetric groups of degrees 16 and 22 |
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Authors: | A. R. Moghaddamfar A. R. Zokayi |
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Affiliation: | 1.Department of Mathematics,Faculty of Science, K. N. Toosi University of Technology, P. O. Box16315-1618, Tehran, Iran;Research Centerfor Complex Systems, K. N. Toosi University of Technology, P. O. Box15875-4416, Tehran, Iran; 2.Department of ElectricalEngineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran; |
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Abstract: | Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p 1, p 2,...,p k }, where p 1 < p 2 < ... < p k , then the degree pattern of G is defined by D(G) = (deg(p 1), deg(p 2),...,deg(p k )). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H| = |G| and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A 22 is OD-characterizable. We also show that the automorphism groups of the alternating groups A 16 and A 22, i.e., the symmetric groups S 16 and S 22 are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected. |
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Keywords: | OD-characterizability of a finite group degree pattern prime graph |
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