2 2 Se p 20 05 The Number of Finite Groups Whose Element Orders is Given
نویسنده
چکیده
For any group G, πe(G) denotes the set of orders of its elements. If Ω is a non-empty subset of N, h(Ω) stands for the number of isomorphism classes of finite groups G such that πe(G) = Ω. We put h(G) = h(πe(G)). In this paper we show that h(P GL(2, p n)) = 1 or ∞, where p = 2 α 3 β + 1 is a prime, α ≥ 0, β ≥ 0 and n ≥ 1. In particular, we show that h(P GL(2, 7)) = h(P GL(2, 3 2)) = ∞.
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ar X iv : m at h / 05 09 50 5 v 2 [ m at h . G R ] 8 O ct 2 00 5 The Number of Finite Groups Whose Element Orders is Given
The spectrum ω(G) of a finite group G is the set of element orders of G. If Ω is a non-empty subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups G with ω(G) = Ω and put h(G) = h(ω(G)). We say that G is recognizable (by spectrum ω(G)) if h(G) = 1. The group G is almost recognizable (resp. nonrecognizable) if 1 < h(G) < ∞ (resp. h(G) = ∞). In t...
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The spectrum ω(G) of a finite group G is the set of element orders of G. If Ω is a non-empty subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups G with ω(G) = Ω and put h(G) = h(ω(G)). We say that G is recognizable (by spectrum ω(G)) if h(G) = 1. The group G is almost recognizable (resp. nonrecognizable) if 1 < h(G) < ∞ (resp. h(G) = ∞). In t...
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تاریخ انتشار 2008