Classical Algebra 2009 First exercise sheet by jasonpeters


									                   Classical Algebra 2009
                    First exercise sheet

Exercise (1.1): Let G be an abelian group. Consider the subsets of G
defined in Chapter 1 of the notes

   • E(G) = {m ∈ G | |m| f inite}

   • Pp (G) = {m ∈ G | m is a p − element}

   • Dp (G) = {m ∈ G | |m|f inite, p |m|}

   • K(G) = {m ∈ G | ∃ a, b ∈ G : m = [a, b]}.

Discuss whether or not these subsets are in fact subgroups of G. (Proof or
counterexample). Give examples (except for K(G)) that they are generally
not subgroups for an arbitrary group G.

Exercise (1.2): Show that the following is a normal subgroup of Aut(G):

       {α ∈ Aut(G)| α(U ) = U f or all abelian subgroups U of G}

Exercise (1.3): Let G be an abelian group. The torsion subgroup GT of
G is discussed in (1C). Show that the torsion subgroup of the abelian group
G/GT has order 1.

Exercise (1.4): Suppose that the center Z(G) of G has order 1. Show for
the centralizer that CAut(G) (Auti (G)) = {1}. (The centralizer is defined in
GT3, Chapter 1.11. )

Exercise (1.5): Let H, K and L be subgroups of the finite group G. Show
that if K ⊆ H then |H : K| ≥ |H ∩ L : K ∩ L|.

Exercise (1.6): Suppose that the finite group G has exactly one maximal
subgroup. Show that G is cyclic.

Exercise (1.7): Suppose that all subgroups H of G, H = G are finite. Is G
necessarily finite?

Exercise (1.8): If p is a prime we have defined a p-solvable group in Chapter
2. Show that if the finite group G is solvable, then it is p-solvable for all
primes p.

Exercise (1.9): Suppose that all proper subgroups of the non-trivial finite
group G are abelian. Show that G contains a normal abelian subgroup
A = {1}.
Hint: This exercise is not very easy.

Exercise (1.10): Let p be a prime and p ≤ n ≤ 2p − 1. Show that the
symmetric group Sn has a p-Sylow complement if and only if n = p.
Hint: Use a theorem in GT3 1.16 or results in Chapter 5 of the group theory

Exercise (1.11): For which subsets π of {2, 3, 5} does the alternating groupp
A5 have a π-Hall subgroup.
Remark: The case π = {3, 5} is mentioned in the notes and need not be
discussed again.

Exercise (1.12): Let k ∈ N. Show that there are only finitely many finite
groups with exactly k conjugacy classes.
Hint: This is not an easy exercise. The proof uses the class equation for a
finite group. You want first to show that an equation on the form
                           1   1         1
                             +   + ... +    =1
                           n1 n2         nk
with ni ∈ N has only finitely many solutions. You also need that there are
only finitely many groups of a given finite order.

Exercise (1.13): Let C be a conjugacy class in the finite group G of order k.
Let |C| = t. Let n ∈ N. Show that the number of elements x ∈ G satisfying
xn ∈ C is divisible by gcd(tn, k).
Conclude that if n | k = |G|, then the number of solutions to the equation
xn = 1 in G is a multiple if n.
Remark: This famous result of Frobenius may be discussed.


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