9. Let G be a group of order 255. Show that (1) Sylow 17-subgroup is normal in G. (ii) 3 a normal subgroup K of order 85. (iii) Kc Z(G).
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- Show that every subgroup of an abelian group is normal.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.4. Prove that the special linear group is a normal subgroup of the general linear group .
- 3. Consider the group under addition. List all the elements of the subgroup, and state its order.Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.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.
- 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.5. Exercise of section shows that is a group under multiplication. a. List the elements of the subgroupof , and state its order. b. List the elements of the subgroupof , and state its order. Exercise 33 of section 3.1. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and is designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.Write 20 as the direct sum of two of its nontrivial subgroups.