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Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order

Recognition of 2-dimensional projective linear groups by the group order and the set of numbers... AbstractIn a finite group G, let πe⁢(G){\pi_{e}(G)}be the set of orders of elements of G, let sk{s_{k}}denote the number of elements of order k in G, for each k∈πe⁢(G){k\in\pi_{e}(G)}, and then let nse⁡(G){\operatorname{nse}(G)}be the unordered set {sk:k∈πe⁢(G)}{\{s_{k}:k\in\pi_{e}(G)\}}.In this paper, it is shown that if |G|=|L2⁢(q)|{\lvert G\rvert=\lvert L_{2}(q)\rvert}and nse⁡(G)=nse⁡(L2⁢(q)){\operatorname{nse}(G)=\operatorname{nse}(L_{2}(q))}for some prime-power q, then G is isomorphic to L2⁢(q){L_{2}(q)}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

Recognition of 2-dimensional projective linear groups by the group order and the set of numbers of its elements of each order

Groups Complexity Cryptology , Volume 10 (2): 8 – Nov 1, 2018

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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1869-6104
eISSN
1869-6104
DOI
10.1515/gcc-2018-0011
Publisher site
See Article on Publisher Site

Abstract

AbstractIn a finite group G, let πe⁢(G){\pi_{e}(G)}be the set of orders of elements of G, let sk{s_{k}}denote the number of elements of order k in G, for each k∈πe⁢(G){k\in\pi_{e}(G)}, and then let nse⁡(G){\operatorname{nse}(G)}be the unordered set {sk:k∈πe⁢(G)}{\{s_{k}:k\in\pi_{e}(G)\}}.In this paper, it is shown that if |G|=|L2⁢(q)|{\lvert G\rvert=\lvert L_{2}(q)\rvert}and nse⁡(G)=nse⁡(L2⁢(q)){\operatorname{nse}(G)=\operatorname{nse}(L_{2}(q))}for some prime-power q, then G is isomorphic to L2⁢(q){L_{2}(q)}.

Journal

Groups Complexity Cryptologyde Gruyter

Published: Nov 1, 2018

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