WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 13, 2014
A Characterization of Some Groups by their Orders and Degree Patterns
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Abstract: Let G be a finite group. Moghaddamfar et al defined the prime graph Γ(G) of group G as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p ∼ q, if there is an element in G of order p · q. Assume $$|G|= p_{1}^{α1}\dotsm p_{k}^{αk}$$ with $$P_{1} <\dotsm< p_{k}$$ and nature numbers $$α_{i}$$ with $$i = 1, 2, · · · , k$$. For $$p ∈ π(G)$$, let the degree of $$p$$ be $$deg(p) = |{q ∈ π(G) | q ∼ p}|$$, and $$D(G) = (deg(p1), deg(p2), · · · , deg(pk))$$. In this note we give an example showing that $$S_{27}$$ is 9-fold $$OD-$$characterizable, which gives a negative answer to an open Problem of Yan et al.
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Keywords: Order component, Element order, Symmetric group, Degree pattern, Prime graph, Simple group
Pages: 586-594
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #57