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Isomorphisms, Automorphisms, Homomorphisms, Kernels

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Isomorphisms, Automorphisms, Homomorphisms, Kernels Isomorphisms, Automorphisms, Homomorphisms, Kernels 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 2 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 fx1; x2; : : : ; xng then its image f(H) is generated by the images ff(x1); f(x2); : : : ; f(xn)g; the image of any cyclic subgroup is cyclic; If f is an isomorphism from G to G0 and x 2 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 2 G consider the mapping from G to itself: cg : G ! G defined by the rule: if x 2 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 f1; x; y; x · yg 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 fx 2 G j f(x) = 1g 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 2 G that commute with every element x 2 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 2 G. Equivalently, if k 2 K then g ·k ·g−1 2 K for all g 2 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 2 G and all h 2 H, we have that g · h · g−1 2 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 2 R) and 1 a12 B1 := all matrices of the form 0 a22 (for a12; a22 2 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. 3.
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