How Abelian is a Finite Group?

A well known theorem of G. A. Miller [4] (see also [2]) shows that a p-group of order p" 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(log n). 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 log 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