1975-16 - Linear and Multilinear Algebra, 1976, Vol. 3, pp....

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Unformatted text preview: Linear and Multilinear Algebra, 1976, Vol. 3, pp. 307-312 CC Gordon and Breach Science Publishers Ltd., 1976 Printed in Great Britain How Abelian is a Finite Group? Dedicated to Olga Taussky P.ERDŐS and E.G . STRAUS University of California at Los Angeles (Received October 7, 1975) INTRODUCTION A well known theorem of G. A. Miller [4] (see also [2])shows that a p-group of orderp" where n > v(v- 1)/2 contains an Abelian subgroup of order p°. It is clear that this theorem together with Sylow's Theorem implies that any finite group of large order contains an Abelian p-group of large order . In this note we use simple number theoretic considerations to make this implication more precise . In Section 1 we show that a group of finite order n contains an Abelian p-group whose order is greater than log n - o(logn). We also give arguments to indicate that the correct answer is probably considerably larger. In the opposite direction it is now known as a result of the work of Adjan and Novikov [1] that Burnside groups with more than one generator whose degree is, say, a sufficiently large prime contain no noncyclic finite or Abelian subgroups . Thus no analogous results about large Abelian subgroups hold for infinite groups. About the upper bounds on the orders of Abelian subgroups of finite groups, it was shown by J. L. Alperin that there exist p-groups of order p3n+2 without Abelian subgroups of order greater than pn +2 . The symmetric group S3, contains no Abelian subgroup of order greater than 3" < N`l log l og N where N = (3n)! = IS3.1 . Thus for any a > 0 there are finite groups G whose largest Abelian subgroup has order o(IGIE) In Section 2 we obtain lower bounds for the number of (ordered) k-tuples of elements of a group G which have pairwise commuting elements . For k = 2 this question was answered by Erdös and Turan [3]. And the general 307 308 P.ERDŐS AND E. G. STRAUS question was raised by Linnik at a conference in Balaton-Füred 1969.As in that paper, the answer is intimately related to the number c(G) of conjugacy classes of G. The growth rate of c(n) = min c(G) n-<IGI <co is very imperfectly understood. The best general result is still ithe one of Landau [3] c(n) > log, log,n....
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1975-16 - Linear and Multilinear Algebra, 1976, Vol. 3, pp....

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