Problem 4. Let G be a group, and suppose that g = G has finite order. Show that k = Z satisfies gk = e if and only if the order |g| of the element g divides k.
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- If a is an element of order m in a group G and ak=e, prove that m divides k.If G is a cyclic group, prove that the equation x2=e has at most two distinct solutions in G.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?
- Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
- 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 .Label each of the following statements as either true or false. Two groups can be isomorphic even though their group operations are different.38. Let be the set of all matrices in that have the form with all three numbers , , and nonzero. Prove or disprove that is a group with respect to multiplication.