By Edward Poon

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**Example text**

The map is a homomorphism of the multiplicative group of non-zero complex numbers into the multiplicative group of non-zero complex numbers (in fact, into the multiplicative group of positive real numbers). Example 3. The map is a homomorphism of the additive group of real numbers into the multiplicative group of positive real numbers. Its inverse map, the logarithm, is also a homomorphism. Let, G, H be groups and suppose H is a direct product Let f: G -+ H be a map, and let J;: G -+ Hi be its i-th coordinate map.

R. Show that H is normal in G. (b) Suppose G is finite. Assume that XiHxi-1 c H for i = 1, ... ,r. Show that H is normal in G. (c) Suppose that H is generated by elements YI"" ,Ym' Assume that Xiyjx i- I e H for all i, j. Assume again that G is finite. Show that H is normal. 18. Let G be the group of Exercise 8 of §1. Let H be the subgroup generated by x, so H = {e, x, X2, X3}. Prove that H is normal. 19. Let G be the group of Exercise 9, §1, that is, G is the quaternion group. Let H be the subgroup generated by i, so H = {e, i, i 2 , i 3 }.

Observe that in additive notation, the condition that S be a set of generators for the group is that every element of the group not 0 can be written Xl where XiES or + ... + X n , -XiES. Example 10. Let G be a group. Let positive integer, we define xn to be X be an element of G. If n is a XX···X, the product being taken n times. If n where m is an integer> 0, we define = 0, we define It is then routinely verified that the rule XO = e. If n = - m [II,§I] GROUPS AND EXAMPLES 21 holds for all integers m, n.