Tutorial 3


1. (i) Given H subset G
	a.When is a H a(inverse) = H ?
	b.What is the no. of distinct copies in collection 
		{ a H a(inverse) | a belonging to G }

  (ii) Stab(t) = g Stab(s) g( inverse)
	where g s.t g(s) = t.


2. Let $G$ act on $S$. Let $C$ be a set of colours. Consider the
action of $G$ on $F(S,C)$, the set of functions from $S$ to $C$. 
Prove the generalization of the Class Equation:
\[ \sum_{O \in Orbits} m(O)=\frac{1}{|G|} \sum_{g\in G} Fixed(g)
\]
where $Fixed(g)=\sum_{f:g(f)=f} mon(f)$. 

3. (i) Compute the cycle polynomial of $D_6 $ acting on the $6$-gon.
(ii) Compute the cycle polynomial of $Z_n $ acting on the regular
$n$-gon. Hint: for a number $m$, let $\phi (m)$ denote the
euler-phi function, i.e., $\phi (m)$ is the number of elements $k$
in the set $\{ 1, \ldots ,m-1\}$ with $gcd(k,m)=1$. 

4. Compute the number of necklaces with seven black and white beads.

5. Let $G$ act on a set $S$. Suppose $p$ is a prime such that $p$
divides $|G|$ but not $|S|$. Show that there is an $s\in S$ such
that $p$ divides $|Stab(s)|$. 

6. Let $G$ be a finite group, and $p$ a prime such that $p$
divides $|G|$. 

Let $S=\{ (g_1 ,\ldots ,g_p )| \prod_{i=1}^p g_i =e \} $. Thus $S$
is the collection of all $p$-tuples of elements of $G$ such that
their product is $e$. Show that $p$ divides $|S|$. 

Let $Z_p $ act on $S$ by circulating the elements of the
$p$-tuple. In other words, for an $i\in Z_p $, and $a=(a_1 ,\ldots
,a_p )\in S$, let $i(a)=(a_{1+i},\ldots ,a_{p+i})$, where all
indices are taken modulo $p$. Show that if $a\in S$, then $i(a)\in
S$. 

What is the cardinality of elements in $S$ which are stabilized by
all elements of $Z_p $? Conclude that there is a $g\in G$ such
that $g^p =e$.