8 Problems for Michaelmas 8.2 Problems for section 2 9 Problems for Epiphany

8.3 Problems for section 3

For these problems, unless otherwise stated, $G$ is a finite group and $H$ is a subgroup of $G$ .

Problem 34.

Let $\rho$ be an irreducible representation of $G$ . Show that $\rho$ is isomorphic to a subrepresentation of a representation induced from an irreducible representation of $H$ .

Solution

Problem 35.

Let $\chi$ be an irreducible character of $H$ and let

\operatorname{Ind}_{H}^{G}\chi=d_{1}\chi_{1}+\ldots+d_{r}\chi_{r}

be decomposition of its induction into irreducible characters of $G$ , with $\chi_{i}$ pairwise distinct and $d_{i}\in\mathbb{Z}_{\geq 0}$ .

Show that

\sum_{i=1}^{r}d_{i}^{2}\leq[G:H].

Solution

Problem 36.

Let $H$ be a subgroup of $G$ . Show that:

1.

If $W_{1},W_{2}$ are representations of $H$ , then

$\operatorname{Ind}_{H}^{G}(W_{1}\oplus W_{2})\cong\operatorname{Ind}_{H}^{G}W_% {1}\oplus\operatorname{Ind}_{H}^{G}W_{2}.$
2.

If $K\subset G$ is a subgroup containing $H$ , and $W$ is a representation of $H$ , then

$\operatorname{Ind}_{H}^{G}W\cong\operatorname{Ind}_{K}^{G}(\operatorname{Ind}_% {H}^{K}W).$
3.

If $V$ is a representation of $G$ , then

$\operatorname{Ind}_{H}^{G}\operatorname{Res}_{H}^{G}V\cong V\otimes% \operatorname{Ind}_{H}^{G}\mathbb{1}.$

Note that all of these may be proved either from the definition (using the ’induction recognition’ corollary) or using characters and Frobenius reciprocity.

Solution

Problem 37.

Let $H\subset G$ be groups and let $\chi$ be a character of $H$ . Let $\dot{\chi}(g)=\chi(g)$ if $g\in H$ and $0$ otherwise.

1.

Show that the formula for the induced character may be rewritten

$(\operatorname{Ind}_{H}^{G}\chi)(g)=\frac{1}{|H|}\sum_{x\in G}\dot{\chi}(x^{-1% }gx).$
2.

If $g_{1}H,\ldots,g_{r}H$ are the left cosets of $H$ in $G$ , show that

$(\operatorname{Ind}_{H}^{G}\chi)(g)=\sum_{i=1}^{r}\dot{\chi}(g_{i}^{-1}gg_{i}).$

Solution

Problem 38.

For each irreducible representation of $S_{4}$ , decompose its induction to $S_{5}$ (where $S_{4}$ is regarded as the subgroup of elements of $S_{5}$ that fix $5\in\{1,\ldots,5\}$ .)

Solution

Problem 39.

Let $\chi$ be the irreducible degree 3 character of $H=A_{5}$ such that

\chi((12345))=\frac{1+\sqrt{5}}{2}.

Let $G=A_{6}$ , with $H$ as the subgroup fixing 6. Compute the character

\operatorname{Ind}_{H}^{G}(\chi).

Solution

Problem 40.

Let $H=\left\langle(12\ldots p)\right\rangle\subset S_{p}=G$ , for $p$ a prime, and $\chi$ be a nontrivial character of $H$ .

1.

Find

$\operatorname{Ind}_{H}^{G}\chi.$
2.

By considering $\left\langle\operatorname{Ind}_{H}^{G}\chi,\operatorname{Ind}_{H}^{G}\chi\right\rangle$ , show that

$(p-1)!\equiv-1\mod p.$

You may use that, if $\zeta$ is a nontrivial $n$ th root of unity, then

1+\zeta+\zeta^{2}+\ldots+\zeta^{n-1}=0.

Solution

Problem 41.

Let

G=\left\langle a,x:a^{7}=x^{3}=e,xax^{-1}=a^{2}\right\rangle.

You may assume that $G$ has 21 elements given by

\{a^{i}x^{j}:i=0,1,\ldots,6,j=0,1,2\}.

1.

Show that $H=\left\langle a\right\rangle$ is a normal subgroup of $G$ .
2.

By considering representations lifted from $G/H$ and induced from $H$ , find the character table of $G$ .

Solution

Problem 42.

Fill in the missing proofs from section 3.6. (This is more of a mega-problem!)

Problems on Mackey theory

The remaining problems in this section concern Mackey theory, which we did not have time to cover and which is therefore not examinable. I leave them here in case you are interested.

Problem 43.

Suppose that $G$ acts transitively on a set $X$ , and that $H\subset G$ is the stabiliser of an element $x_{0}\in X$ .

Find a bijection between $H\backslash G/H$ and the orbits of $G$ on the product $X\times X$ (with the action $g(x,y)=(gx,gy)$ ).

Solution

Problem 44.

For the following pairs of groups $H\subset G$ , find a set of double coset representatives for $H$ in $G$ .

1.

$H=\{e,s\}\subset G=D_{5}$
2.

$H=\{e,(123),(132)\}\subset G=S_{4}$
3.

$H\cong S_{3}\times S_{2}\subset S_{5}$ the subgroup of elements $\sigma$ such that $\sigma$ preserves the subsets $\{1,2,3\}$ and $\{4,5\}$ .

Solution

Problem 45.

Prove Lemma 3.19: if $H$ and $K$ are subgroups of $G$ and $s\in G$ , then

HsK=h_{1}sK\sqcup\ldots\sqcup h_{r}sK

where $h_{1},\ldots,h_{r}$ are left coset representatives for

H_{s}=H\cap sKs^{-1}

in $H$ .

Solution

Problem 46.

Suppose that $H, K$ are subgroups of $G$ and that their orders are coprime. Suppose that $\rho$ is a representation of $H$ and that $\sigma$ is a representation of $K$ . Show that

\dim\operatorname{Hom}_{G}(\operatorname{Ind}_{H}^{G}\rho,\operatorname{Ind}_{% K}^{G}\sigma)=\dim(\rho)\dim(\sigma)\frac{|G|}{|H||K|}.

Solution

Problem 47.

Suppose that $H\subset G$ is a subgroup of index two, that $\epsilon:G\to\mathbb{C}^{\times}$ is the nontrivial character with kernel $H$ , and that $s\in G$ is not in $H$ .

Show that, if $\sigma$ is an representation of $G$ , then $\operatorname{Res}^{G}_{H}\sigma$ is irreducible if and only if $\sigma\not\cong\epsilon\sigma$ . Moreover, show that if $\sigma\cong\epsilon\sigma$ , then

\operatorname{Res}^{G}_{H}\sigma\cong\rho\oplus{}^{s}\rho

for $\rho$ an irreducible representation of $H$ with $\rho\not\cong{}^{s}\rho$ .

Solution

Problem 48.

Suppose that $\chi$ is an irreducible character of $S_{n}$ whose degree $\chi(e)$ is odd. Show that there exists an odd permutation $g\in S_{n}$ such that $\chi(g)\neq 0$ .

Hint: use the previous problem.

Solution