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HOMEWORK 2
ALGEBRA-I, MATH 5613, FALL 2009
Exercise 1. Let N be a normal subgroup of a finite group G, such that the orders of N and G/N are relatively prime.
(1) Let H be a subgroup of G of the same order as G/N. Show that G = HN. (2) For any φ ∈ Aut(G), show that φ(N ) = N.
Exercise 2. Let H, H 1 and H 2 be subgroups of finite index in G.
(1) Show that H 1 ∩ H 2 has finite index in G. (2) Show that there is a normal subgroup N of finite index in G, such that N ⊂ H.
Exercise 3. An automorphism φ : G → G is said to be an inner automorphism if there is g ∈ G such that φ(x) = gxg−^1 for all x ∈ G. Let Int(G) denote the set of all inner automorphisms of G.
(1) Show that Int(G) ' G/Z, where Z is the center of G. (2) Show that Int(G) is a normal subgroup of Aut(G).
Exercise 4. Describe Hom(Dn, C∗). Recall that Hom(Dn, C∗) stands for the set of all homomorphisms from the dihedral group Dn to nonzero complex numbers.
Exercise 5. (Centers of Sn and An)
(1) If n ≥ 3, then show that the center of Sn is trivial. (2) If n ≥ 4, then show that the center of An is trivial.
Exercise 6. The Klein 4-group Z/ 2 Z × Z/ 2 Z is denoted V 4.
(1) Show that A 4 sits in an exact sequence: 0 −→ V 4 −→ A 4 −→ Z/ 3 Z −→ 0. (2) Show that A 4 does not have a subgroup of order 6. (3) Draw the lattice of subgroups of A 4. (4) Construct a nonabelian group of order 12 not isomorphic to A 4.
Exercise 7. Show that the following sets are generating sets for Sn.
(1) {(1, 2), (1, 3),... , (1, n)}. (2) {(1, 2), (2, 3),... , (n − 1 , n)}. 1
2
(3) {(1, 2), (1, 2 ,... , n)}. (4) If n = p a prime, then {(r, s), (1, 2 ,... , p)}. Here (r, s) means any 2-cycle.
(The last one will be useful in Galois theory to construct a Galois extension over Q with Galois group Sp–this is part of the so-called inverse Galois problem.)
Let (Z/nZ)×^ denote the multiplicative group of integers modulo n, which are rela- tively prime to n; the group law is multiplication modulo n. (You should check for yourself that this makes sense and is indeed a group. To see if you understand it, convince yourself of the following examples: (Z/ 4 Z)×^ ' Z/ 2 Z, (Z/ 5 Z)×^ ' Z/ 4 Z, (Z/ 16 Z)×^ ' Z/ 2 Z × Z/ 4 Z.)
Exercise 8. Recall that Cn is a cyclic group of order n. We denote the greatest common divisor of two integers a and b by (a, b). Let G be a group and g ∈ G.
(1) If o(g) = n, then o(gi) = n/(i, n). (2) Given o(g) = n, show that o(gi) = n if and only if (i, n) = 1. (3) Let ψ ∈ Aut(Cn). Show that there is an integer iψ, which is relatively prime to n, and such that ψ(x) = xiψ^ for all x ∈ Cn. (4) Show that the map ψ 7 → iψ induces an isomorphism Aut(Cn) ' (Z/nZ)×.
Exercise 9. Let F be a field. Consider the map T : F ∗^ → Aut(F ) given by x 7 → Tx, where Tx(y) = xy for all x, y ∈ F. Consider the corresponding semidirect product F o F ∗. Show that
F o F ∗^ '
a b 0 1
: a ∈ F ∗, b ∈ F
where the right hand side is viewed as a subgroup of GL 2 (F ).
(Some mathematicians, especially those who work in a subject called Automorphic forms, call this subgroup as the mirabolic subgroup of GL 2 (F ); this terminology is attributed to Herv´e Jacquet.)
