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Math 210A Homework 3
Question 1. Let C be a category, I a small category and F : I → C a functor. (a) Prove that lim i∈I F (i) is characterized by the existence, for every T ∈ C, of a natural bijection (of sets) MorC (T, lim i∈I F (i)) ∼= lim i∈I MorC (T, F (i)). (b) Similarly, characterize colim i∈I F (i) as follows : For every T ∈ C, there is a natural bijection MorC (colim i∈I F (i) , T ) ∼= lim i∈I MorC (F (i), T ).
Question 2. Let A be a semigroup (that is, a set with an associative law a · b)
(a) Suppose A has a left identity element eL ∈ A (that is, eL · a = a for each a ∈ A). Suppose further that each element a ∈ A has a left inverse. Prove that A is a group. (b) Suppose now that A has a left identity and every element has a right inverse. Is this enough to conclude that A is a group?
Question 3. Let G be a cyclic group. (a) Describe all subgroups of G. (b) Find all automorphisms of G. Question 4.
(a) Show that if g^2 = e for every g in a group G, then G is abelian. (b) Show that every subgroup of index p = 2 is normal. (c) Let p be an odd prime. Find a group with a non-normal subgroup of index p. (d) Prove that if G is a finite group of even order, then G contains an element a such that a 6 = e but a^2 = e.
Question 5.
(a) Determine the order of the symmetric group Sn. (b) Prove that Sn is generated by all the transpositions. (c) Prove that Sn is, in fact, generated by the transpositions (1, 2), (1, 3), ..., (1, n). (d) Prove that Sn can be generated by the transposition (1, 2) and the n-cycle (1, 2 , ..., n).
Question 6. Let D 8 be the group of isometries of a square (distance-preserving bijections).
(a) Show that it is generated by two elements ρ and σ such that ρ^4 = 1, σ^2 = 1 and σρ σ = ρ−^1. (b) Determine all subgroups of D 8. (c) Find subgroups K C H C D 8 such that K is not normal in D 8.
Question 7. Let n ≥ 1. Define a group by generators and relations as D 2 n = 〈 ρ, σ | ρn^ = σ^2 = σρ σρ = 1〉. It is called the dihedral group of order 2 n.
(a) Show that D 2 n indeed has order 2n. (Hint: Embed D 2 n into EndAb(C).) (b) Identify D 2 n as the isometries of the regular n-gone (n ≥ 3). (c) Determine the center Z(D 2 n). (d) Find all normal subgroups of D 2 n. (e) Prove that D 6 ∼= S 3 , but that D 8 6 ∼= S 4.
Question 8. Let Inn(G) ⊂ Aut(G) be the subgroup of inner automorphisms of G (that is, automorphisms of the form a 7 → gag−^1 for some g ∈ G). Prove that Inn(G) is a normal subgroup of Aut(G). Question 9.
(a) Let G be a group, and let N be a subgroup of the center Z(G). Show that N is normal in G. Prove that if G/N is cyclic then G is abelian. (b) Let G be a group and suppose Aut(G) is cyclic. Prove that G is abelian. (Hint: Use the group Inn(G), defined in Question 8 and compare with part (a) above for N maximal.)
Question 10. Show that a group with no non-trivial automorphism is trivial or isomorphic to Z/ 2 Z. Hint: First check it is abelian and 2-torsion.
