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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 $qp1$. Let $A$ be a nonAbelian 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 nonAbelian simple group acted on nontrivially by another finite nonAbelian 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 $p^{n}p^{n1}$. These are in bijection with monic polynomials of degree $n$ with nonzero 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 twiceodd order have quaternionic representations?
typo 
Nov 12 
answered  Can groups of twiceodd 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 $q = p$. 