(b) Let H= ((3,3, 6)), the cyclic subgroup of G generated by (3,3,6). Determine |G/H.
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- 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?Let H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is a conjugate of H and that H and gHg1 are conjugate subgroups. Prove that H is abelian, then gHg1 is abelian. Prove that if H is cyclic, then gHg1 is cyclic. Prove that H and gHg1 are isomorphic.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.
- 3. Consider the group under addition. List all the elements of the subgroup, and state its order.Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.Let H be a torsion subgroup of an abelian group G. That is, H is the set of all elements of finite order in G. Prove that H is normal in G.
- Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .Find an isomorphism from the additive group to the multiplicative group H={ [ 1n01 ]n } and prove that (x+y)=(x)(y). Sec. 3.4,14 Prove that the set H={ [ 1n01 ]n } is cyclic subgroup of the group GL(2,).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.