10. Let G be a finite group and a € G. Use Lagrange's Theorem to prove that the order of a divides the order of G.

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Answered: 10. Let G be a finite group and a € G.…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 30E: Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G...

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Exercises 35. Prove that any two groups of order are isomorphic.
Find two groups of order 6 that are not isomorphic.
45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )
Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.
Let H and K be arbitrary groups and let HK denotes the Cartesian product of H and K: HK=(h,k)hHandkK Equality in HK is defined by (h,k)=(h,k) if and only if h=h and k=k. Multiplication in HK is defined by (h1,k1)(h2,k2)=(h1h2,k1k2). Prove that HK is a group. This group is called the external direct product of H and K. Suppose that e1 and e2 are the identity elements of H and K, respectively. Show that H=(h,e2)hH is a normal subgroup of HK that is isomorphic to H and, similarly, that K=(e1,k)kK is a normal subgroup isomorphic to K. Prove that HK/H is isomorphic to K and that HK/K is isomorphic to H.
24. Assume that the group is a homomorphic image of the group . Prove that is cyclic if is cyclic. Prove that divides , whether is cyclic or not.
Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
For a fixed group G, prove that the set of all automorphisms of G forms a group with respect to mapping composition.

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