If G is a group and H is a subgroup of index 2 in G show that H is a normal subgroup of G.
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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?18. If is a subgroup of , and is a normal subgroup of , prove that .4. Prove that the special linear group is a normal subgroup of the general linear group .
- 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=4018. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.