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3h
comment Can we say that $A$ is a complement for a group $G$?
Take $n = 1 + pq,$ where $p,q$ are primes with $q|p-1$. Let $A$ be a non-Abelian group of order $pq.$ Then $A$ is not a Frobenius complement, but $A$ does have a (complex) character which takes value $n$ on the identity and $1$ everywhere else. The character in question is the sum of the trivial character and the regular character.
9h
revised Groups with a unique composition series
typos
1d
awarded  Nice Answer
2d
revised Groups with a unique composition series
typo
2d
revised Groups with a unique composition series
Clarified
2d
comment Groups with a unique composition series
@spin: Note that the (now proved) Schreir conjecture tells us that you can't have an extension of a finite non-Abelian simple group acted on non-trivially by another finite non-Abelian simple group.
2d
comment Groups with a unique composition series
$F(G)$ is the largest nilpotent normal subgroup of the finite group $G$, the Fitting subgroup of $G$. Also, $F^{\ast}(G)$ is the generalized Fitting subgroup of $G$.
2d
revised Groups with a unique composition series
Typo
2d
answered Groups with a unique composition series
Nov
21
revised A question on the representation theory of finite group
clarified
Nov
20
comment irreducible representation of a simple Lie group where each element has a fixed point
What about the derived group of ${\rm SO}(3,\mathbb{R})?" (in its natural representation).
Nov
20
answered A question on the representation theory of finite group
Nov
20
comment A question on the representation theory of finite group
@AlexDegtyarev : I interpret the question as asking whether, given $k$ (not necessarily distinct), integers $m_{1},m_{2}, \ldots, m_{k}$ with $\sum_{i=1}^{k}m_{k}^{2} = n,$ can we finite group $G$ such that the $m_{i}$ are the degrees of the complex irreducible characters of $G$ (including multiplicities). I think that question is actually quite difficult, if not impossible, to answer fully, though there are many necessary conditions.
Nov
20
revised Normal Covering of a Finite Group
added extra comment
Nov
17
comment Representation theory of the general linear group over a finite prime field
The number of such inequivalent irreducible modules is well known to be $p^{n}-p^{n-1}$. These are in bijection with monic polynomials of degree $n$ with non-zero constant term in $\mathbb{F}_{p}[x].$ However, explicit description of the simple modules is, I believe, notoriously difficult.
Nov
12
revised Can groups of twice-odd order have quaternionic representations?
typo
Nov
12
answered Can groups of twice-odd order have quaternionic representations?
Nov
9
revised Lucido's three prime lemma
added clarification
Nov
8
revised Lucido's three prime lemma
discussed the structure of groups in which all elements have prime power order.
Nov
5
revised Hall subgroups of general linear group
added comment on case $q = p$.