# 14, cannot use Sylow\'s theorem, must use only definitons and results about no

Question

# 14, cannot use Sylow's theorem, must use only definitons and results about normal subgroups and subgroups

Find groups H and G such that H subset G subset A_4 and the following conditions are satisfied: a. H is a normal subgroup of G. b. G is a normal subgroup of A_4. c. H is not a normal subgroup of A_4. (Thus the statement "A normal subgroup of a normal subgroup is a normal subgroup" is false.) Find groups H and K such that the following conditions are satisfied: a. H is a normal subgroup of K. b. K is a normal subgroup of the octic group. c. H is not a normal subgroup of the octic group.

Explanation / Answer

H = { () , (1,2) , (3,4) } is a normal subgroup of G.

Now , we have to find a group G such that h G = G h for all h in H.

We know the klein four group which is normal subgroup of A4.

V4 = {() , (1,2 ) (3,4) , (1,3 ) (2,4) , (1,4) (2,3) }

Thus, G is V4.

answer.

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