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OD-Characterization of alternating and symmetric
groups of degrees 16 and 22 |
| A. R. MOGHADDAMFAR1,A. R. ZOKAYI2, |
| 1.Department of Mathematics,
Faculty of Science, K. N. Toosi University of Technology, P. O. Box
16315-1618, Tehran, Iran；Research Center
for Complex Systems, K. N. Toosi University of Technology, P. O. Box
15875-4416, Tehran, Iran; 2.Department of Electrical
Engineering, 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) = {p1, p2, …, pk}, where p1<p2<…<pk, then the degree pattern of G is defined by D(G) = (deg(p1), deg(p2), … , deg(pk)). 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 Odcharacterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A22 is OD-characterizable. We also show that the automorphism groups of the alternating groups A16 and A22, i.e., the symmetric groups S16 and S22 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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Issue Date: 05 December 2009
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