Isomorphisms, Automorphisms, Homomorphisms, Kernels

Home , Automorphism, Group isomorphism, Homomorphism, Isomorphism

September 10, 2009

Deﬁnition 1 Let G, G0 be groups and let f : G → G0 be a mapping (of the underlying sets). We say that f is a (group) homomorphism if f(x · y) = f(x) · f(y) for all x, y ∈ G.

The composition f G−→G0−→h G” of two homomorphisms (as displayed) is again a homomorphism. For any two groups G, G0 by the trivial homomorphism from G to G0 we mean the mapping G → G0 sending all elements to the identity element in G0. Recall:

Deﬁnition 2 A group homomorphism f : G → G0 is called a group isomorphism—or, for short, an isomorphism—if f is bijective. An isomorphism from a group G to itself is called an automorphism.

Exercise 1 Let f : G → G0 be a homomorphism of groups. The image under the homomorphism 0 f of any subgroup of G is a subgroup of G ; if H ⊂ G is generated by elements {x1, x2, . . . , xn} then its image f(H) is generated by the images {f(x1), f(x2), . . . , f(xn)}; the image of any cyclic subgroup is cyclic; If f is an isomorphism from G to G0 and x ∈ G then the order of x (which—by deﬁnition—is the order of the cyclic subgroup generated by x) is equal to the order of f(x).

For G a group, let Aut(G) be the set of all automorphisms of G, with ‘composition of automorphisms” as its ‘composition law’ (denoted by a center-dot (·).

As noted in class,

Proposition 1 If G is a group then Aut(G) is also a group.

1 Exercise 2 1. Let G be a group. For any g ∈ G consider the mapping from G to itself:

cg : G → G

deﬁned by the rule: if x ∈ G then

−1 cg(x) := g · x · g .

Show that cg : G → G is an automorphism of the group G. (It is called conjugation by g). 2. Show that the assignment g 7→ cg deﬁnes a homomorphism of groups,

c : G → Aut(G).

Note that G is commutative if and only if c is the trivial homomorphism.

Exercise 3 1. Let G be a group of order three. Show that G is cyclic. Show that Aut(G) is of order two.

2. Let G be a group of order 5. Show that G is cyclic. Show that Aut(G) is cyclic of order 4.

Exercise 4 Let G be the set with composition law denoted by center-dot (·) consisting of four elements {1, x, y, x · y} where every element is its own inverse. Show that the previous sentence does indeed describe a group. Now give a complete description of Aut(G).

Deﬁnition 3 Let f : G → G0 be a group homomorphism. By the kernel of f we mean the subset of G consisting of the set of elements {x ∈ G | f(x) = 1}

Exercise 5 1. Let f : G → G0 be a group homomorphism. Show that the kernel of f is a subgroup of G.

2. Let G be a group. Show that the kernel of the homomorphism c : G → Aut(G) deﬁned in the previous exercise consists of the set of all elements g ∈ G that commute with every element x ∈ G. It is called the center of G.

3. Let f : G → G0 be a group homomorphism, and let K ⊂ G be the kernel of f. Show that K is preserved under conjugation by any element g ∈ G. Equivalently, if k ∈ K then g ·k ·g−1 ∈ K for all g ∈ G.

2 Deﬁnition 4 We will say that a subgroup H ⊂ G is normal in G if it has the property you have just proved holds for “kernels” in the previous exercise; that is: H ⊂ G is normal in G if for all g ∈ G and all h ∈ H, we have that g · h · g−1 ∈ H, or equivalently, g · H · g−1 = H.

We will see later that this property of normality characterizes kernels: any normal subgroup of G is the kernel of some homomorphism—but that will be for later.

Exercise 6 1. Show—really just note— that the center of any group is normal in that group. 2. If N ⊂ G is a normal subgroup in G and H ⊂ G is a subgroup, is N ∩ H a normal subgroup of H?

3. Consider the subsets in GL2(R) given by

a a B := all matrices of the form 11 12 0 a22

(for a11, a12, a22 ∈ R) and

1 a12 B1 := all matrices of the form 0 a22

(for a12, a22 ∈ R) so that B1 ⊂ B ⊂ GL2(R).

Show that B1 is a normal subgroup of B, but neither B1 nor B are normal in GL2(R).

Exercise 7 Let G = D6 be the group of symmetries of the equilateral triangle. Find the center of G, describe all normal subgroups of G, compute Aut(G) and give a complete description of the homomorphism c : G → Aut(G) of the previous exercise.

Exercise 8 Do the same as above for Let G = D10, the group of symmetries of the regular pentagon. Find the center of G, describe all normal subgroups of G, compute Aut(G) and give a complete description of the homomorphism c : G → Aut(G) of the previous exercise.