Math 517A - Group Theory I

Fall 2014 - Section 1
Schedule: TTh 11 AM to 12.15 PM PAS 416. Instructor: K. Lux

Text: We will use the book ''Groups and Characters'' by L. Grove.

We will focus mainly on the chapters dealing with representation theory and characters of finite groups over fields not dividing the order of the group.

1st homework: Do the exercises in Section 5.1 of the book. Due date: September 11, in class.

2nd homework: Due date: September 18, in class.

  1. Write a GAP program that implements Maschke's theorem (given matrices for generators of the group and the order) and use it to decompose the natural permutation representation of the symmetric group on 5 and 6 letters over the field with 7 elements. Note: You may assume that you are given matrices for a generating set of the group in upper blockdiagonal form, and that the list of the blocksizes are also part of the input.
  2. Let $\hat{T}_1,\ldots,\hat{T}_r$ be the irreducible matrix representations (up to equivalence) of a finite group $G$ over a splitting field $F$ not dividing the order of $G$. Define for $k=1,\ldots,r$ the following elements in $FG$, the group algebra: $p^k_{ij} = \frac{deg(\hat{T}_k)}{|G|}\sum_{g \in G} (\hat{T_k})_{ji}(g^{-1})g$ in $FG$. Show that the $p^k_{ij}$ are a basis of $FG$ and give a formula for the product of two basis elements as a linear combination in terms of the basis.
  3. Let $\hat{T}$ be an absolutely irreducible matrix representation of a group $G$ of degree $n$ over a field $F$ not dividing the order of $G$. Show that $\hat{T}(G)$ spans $M_{n \times n}(F)$.
3rd homework: Due date September, 25, in class:
  1. Show that $Z(FG)$, the center of a group algebra $FG$, is a a subalgebra of $FG$ and that the conjugacy class sums are an $F$-basis of $Z(FG)$. Furthermore show that the the number$n_{ijk}$ defined in class are the structure constants of $Z(FG)$ with respect to the conjugacy class sum basis.
  2. Check that Wedderburn's theorem from the core course Math 511A/B can be applied to $FG$, where $G$ is a finite group and $F$ is a field, whose characteristic does not divide the order of $G$. Use Wedderburn's theorem to determine the dimension of $Z(FG)$, when $F$ is a splitting field for $G$, and use that to show that the number of conjugacy classes of $G$ is the number of irreducible $F$-characters of $G$.
  3. Let the finite group $G$ act on the finite set $X$ and let $\pi_X$ be the (complex) permutation character. Recall the Cauchy-Frobenius-Burnside theorem and give an interpretation in terms of $\pi_X$.
4th homework: Due date October, 7, in class:
  1. Let $T \colon G \rightarrow {\rm GL}_n(\mathbb{C})$ be a complex matrix representation of the finite group $G$. Show that $T$ is equivalent to a unitary representation $ T^\prime \colon G \rightarrow {\rm U}_n(\mathbb{C})$. Show that if $T,T^\prime \colon G \rightarrow {\rm U}_n(\mathbb{C})$ are equivalent representations, then they are unitarily equivalent.
  2. Let $G$ be a finite group, $K_1$,$K_2$,$K_3$ three conjugacy class of $G$ and let $x_1 \in K_1$,$x_2 \in K_2$ and $x_3 \in K_3$. Show that the structure constant $n_{123}$ for $Z(\mathbb{C}G)$ is determined by the character table of $G$ by showing: $n_{123}= \frac{|G|}{|C_g(x_1)| |C_G(x_2)|} \sum_{\chi \in {\rm Irr}(G)} \frac{\chi(x_1)\chi(x_2)\overline{\chi(x_3)}}{\chi(1)} $.
  3. Continuation of the Problem 3 on the 2nd homework. Let $T \cong \bigoplus_{k=1}^r n_kT_k$ be a representation of $G$ and let $V$ be the corresponding $FG$-module. Define for $k=1,\ldots,r$ the following elements in $FG$: $q^k = \frac{deg(\hat{T}_k)}{|G|}\sum_{g \in G} \chi_{\hat{T}_k}(g^{-1})g$ in $FG$. Set $V_k:=q^k(V)$. Show that $V$ is the direct sum of the $FG$-submodules $V_k$ for $k=1,\dots,r$ and show that the representation of $G$ on $V_k$ is equivalent to $n_kT_k$.
5th homework: Due date October, 23, in class:
  1. Do the Exercises page 119 and page 125.
  2. Let $W$ be an $FG$-module and suppose $W$ is the internal direct sum of the $F$-subspaces $W_1,\dots,W_n$ that are permuted transitively by $G$ (via its action on $W$). Let $H$ be the stabilizer of $W_1$ that means $H=\{ g \in G| g(W_1)=W_1\}$. Show that the $FG$-module $W$ is isomorphic to an $FH$-module $V$ induced up to $G$.
    1. Let $G$ be a finite group and let $g \in G$. Show that $g$ is conjugate to the commutator $[x,y]$ of $x,y \in G$ if \[ \sum_{\chi \in \text{Irr}(G)} \frac{|\chi(x)|^2\overline{\chi(g)}}{\chi(1)} \ne 0. \]
    2. Show that $g$ is a commutator in $G$ if \[ \sum_{\chi \in \text{Irr}(G)} \frac{\chi(g)}{\chi(1)} \ne 0. \]
6th homework due November, 25th, in class:
  1. Do the exercises page 141,142, and 147 in the book.
  2. For the group $3.A_6$ use its character table in GAP to show:
    1. $3.A_6$ is not a simple group,
    2. it is equal to its commutator subgroup,
    3. There are elements in $3.A_6$ that are not commutators.
    Note that the Ore conjecture says this cannot happen for a nonabelian simple group. This conjecture is now a theorem proven by Pham Tiep et al.
K. Lux / Department of Mathematics / University of Arizona / last revised November, 12, 2014