VIEWS: 14 PAGES: 2 CATEGORY: Financial Planning & Retirement POSTED ON: 5/19/2009 Public Domain
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? 1 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 notes. 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. 2