4. Let G be a finite group of order 4 containing no element of order 4. Explain why every nonidentity element of G has order 2
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- Suppose 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.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?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 .
- 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.If a is an element of order m in a group G and ak=e, prove that m divides k.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
- 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.
- Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.34. Suppose that and are subgroups of the group . Prove that is a subgroup of .12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.