Orders for Which a Given Number of Groups Exist
نویسندگان
چکیده
منابع مشابه
Orders for Which There Exist Exactly Two Groups
Introduction. A long-standing problem in group theory is to determine the number of non-isomorphic groups of a given order. The inverse problem–determining the orders for which there are a given number of groups–has received considerably less attention. In this note, we will give a characterization of those positive integers n for which there exist exactly 2 distinct groups of order n (up to is...
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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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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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ژورنال
عنوان ژورنال: Proceedings of the National Academy of Sciences
سال: 1932
ISSN: 0027-8424,1091-6490
DOI: 10.1073/pnas.18.6.472