Counting classes and characters of groups of prime exponent. (Q2472748)

This is a very careful analysis of character degrees and conjugacy class sizes of some classes of \(p\)-groups including the frequencies with which each occurs. The \(p\)-groups are relatively free groups of given class, sometimes with added conditions. In particular they completely determine the data for conjugacy classes for nilpotency class up to four and exponent \(p\). However the character data is only complete for nilpotency class up to three. As a flavour of the results we quote Theorem 5: Given a relatively free group of rank \(r\), exponent \(p\) and nilpotency class \(2\) the character degrees are \(1,p,p^2,\dots,p^{[r/2]}\). If \(0<2k\leq r\), then the number of characters of degree \(p^k\) is \[ \frac{p^{r+k^2-3k}(p^r-1)(p^{r-1}-1)\cdots(p^{r-2k+1}-1)}{(p^{2k}-1)(p^{2k-2}-1)\cdots (p^2-1)}. \] As another example we give Lemma 9: Let \(L\) be the free group of rank \(3\), exponent \(3\) and class \(3\). Then class sizes are \((1,3,27)\) with multiplicities \((3,26,78)\) and the character degrees are \((1,3,27)\) with multiplicities \((27,78,2)\). -- The paper is well written and worth reading.