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If $g\in G$ has order $n$, then its eigenvalues are all $n$th roots of unity. So if $V$ is a representation with degree greater than $n$, the eigenvalues of $g$ on $V$ can't all be distinct. There ar …

Let $G=(\mathbb{Z}/p\mathbb{Z})^n$, for $p$ a prime and $n>1$. Then $l(G)=n$ but $\Lambda(G)$ can be made as large as you like by choosing $p$ large.

As Geoff thought, the answer is contained in James and Kerber (it's Theorem 2.5.13 in "The Representation Theory of the Symmetric Group", Encyclopedia of Mathematics and its Applications vol. 16, 1981 …

How about $D_{n(n-1)}\times C_{n-2}\times S_{n-3}$ for any $n$ where 3 divides $n(n-1)$ but 4 doesn't?

Let $G=(C_7\rtimes C_3)\times(C_5\rtimes C_2)$. Then $G$ is supersolvable but doesn't have a normal subgroup of order $7\times 2$ or $3\times 5$.

Let $k$ be a field of characteristic $2$, and let $A$ be the path algebra over $k$ of the quiver with two vertices, $v_1$ and $v_2$, and arrows $a:v_1\to v_2$ and $b:v_2\to v_1$, modulo the relations …

No. For example, take the direct product of a dihedral group of order $2p$ and a cyclic group of order $(p^2+1)/2$.

I'm supervising an undergraduate research project. Among other things, I've got the student to look at this paper of Gene Kopp and John Wiltshire-Gordon. This question arose from a missing complex con …

No. There are non-abelian groups $G$ for which all subgroups are normal, such as the quaternion group of order 8. So the intersection of all normalizers is just $G$.

Here's a (sketch of a) proof that this cohomology group is always zero using the fact that $G=GL_2(\mathbb{F}_p)$ has a cyclic Sylow $p$-subgroup (and so it definitely doesn't generalize easily to $GL …

Let $G=C_2\times C_2$ and $H$ a subgroup of order $2$. Let $U$ be the non-trivial irreducible module on which $H$ acts trivially, and $V$ one of the other non-trivial irreducible modules. Then $U\oti …

There are $70$ homomorphisms from $\mathbb{Z}\times\mathbb{Z}$ to the dihedral group of order $14$.

This won't be true if $G=\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$. If $G$ has a local ring structure with maximal ideal $\mathfrak{m}$, and quotient field $G/\mathfrak{m}$ isomorphic to $\ …

If a finite group has a presentation with $g$ generators and $r$ relations, then the Schur multiplier is generated by $r-g$ elements. There's been lots of work studying groups and presentations where …

I think this example answers both questions. Let $k$ have characteristic $3$, and let $G=C_3\times S_3$. Then $kG$ has two simple modules, both one-dimensional, and for each simple module $S$, $\tex …