Timeline for Characters of p-groups

6 events

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Jun 28, 2016 at 11:19	comment	added	yakov		Stanley has proved that the number of nonlinear irreducibles for a nonabelian $p$-group is a multiple of $p-1$. But Mann asserts more: the number of irreducibles of a given degree $>1$ is a multiple of $p-1$.
Mar 19, 2013 at 13:00	comment	added	Richard Stanley		In fact, if $M$ is the number of nonlinear irreducible characters, then $M\equiv 0$ (mod $p^2-1$) if $n\equiv r$ (mod 2); $M\equiv p-1$ (mod $p^2-1$) if $n$ is odd and $r$ is even; and $M\equiv -p+1$ (mod $p^2-1$) if $n$ is even and $r$ is odd.
Mar 19, 2013 at 12:21	vote	accept	Amin
Mar 18, 2013 at 7:57	comment	added	Geoff Robinson		@Richard,@KConrad: And Richard's answer and/or KCs comment makes the situation with the conjugacy class question transparent too: using the modified class equation gives $1 \equiv 1 + (k(P) -\|Z(P)\|)$ (mod $p-1$), where $k(P)$ is the number of conjugacy classes of $P$, so the number of conjugacy classes of $p$ of size greater than $1$ is diisible by P-1$. Simpler than my approach!
Mar 18, 2013 at 7:07	comment	added	KConrad		It's easier than that: letting $N$ be the number of terms in that sum on the right, just look at this equation mod $p-1$. Since $p \equiv 1 \bmod p-1$, the equation becomes $1 \equiv 1 + N \bmod p-1$, so $N \equiv 0 \bmod p-1$.
Mar 18, 2013 at 0:33	history	answered	Richard Stanley	CC BY-SA 3.0