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Exercises 1. Show that intersection of two subgroups of a group G is a subgroup of G. 2. Let G be the Quaternion group. Find centre of G. Find also the normalizer of i in G. 3. Show that a group cannot be written as union of two (proper) subgroups, although it is possible to express it as union of three subgroups. 4. If H is a subgroup of G, show that g Hg = {g hg |he H; is a subgroup of G. Show further that g Hg is abelian if H is abelian. 5. Show that U as defined in example 20 on page 64, is a subgroup of U. 6. Let G be a finite abelian group under addition and let n e Z be a fixed positive integer. Show that nG = {nx ( x e G; and G[n] = {x e G |nx = 0} are subgroups of G, where 0 is identity of G. (See problem 48 on page 263). 7. If G is a group of order 91, show that it cannot have two subgroups of order 13. 8. If H c K are two subgroups of a finite group G then show that i G(H) = ic(K) ix(H). 9. Show that normalizer of an element a in a group G is a subgroups of G. 10. Show that H = {0, 2, 4} is a subgroup of Zo = 10, 1, 2, 3, 4, 5} addition modulo 6. 11. Let G be the group of all 3 x 3 invertible matrices over reals. Show that 1 a b H = 0 1 c a,b, cER) is a subgroup of G. 0 0 12. If H and K are subgroups whose orders are relatively prime then show that H nK = (e)