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Recognizing \(L_3(4)\) by the set of element orders in the class of all groups. (English. Russian original) Zbl 1333.20039

Algebra Logic 54, No. 4, 279-282 (2015); translation from Algebra Logika 54, No. 4, 439-443 (2015).
From the introduction: The spectrum of a periodic group \(G\) is the set \(\omega(G)\) of its element orders. The objective of the present paper is to consider the simple group \(L_3(4)\) which is isomorphic to \(M_{21}\).
Theorem. If \(\omega(G)=\{1,2,3,4,5,7\}=\omega(L_3(4))\), then \(G\) is isomorphic to \(L_3(4)\).
For finite groups \(G\), the result was proved by W. Shi [in J. Math., Wuhan Univ. 5, 191-200 (1985; Zbl 0597.20007)].

MSC:

20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D06 Simple groups: alternating groups and groups of Lie type

Citations:

Zbl 0597.20007

Software:

GAP
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Full Text: DOI

References:

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