When G be a group and Let H contain e along with the set of all elements in G of

Question

When G be a group and Let H contain e along with the set of all elements in G of order 2. a) Prove that H is a subgroup if and only if G is abelian. b) Find a counterexample of this for a non-abelian group. Please show all steps!!!!

Explanation / Answer

a. to show that its a subgroup, we need to see that a, b in H => ab in H (ab)^2 = a^2 b^2 (iff the group is abelian) = e e = e if a in H then a^-1 in H -> a^2 = e so, (a^-1)^2 = a^-2 = e as a^2 a^ -2 = e So, this does the first part. b. Counter example, consider the group of quarternions - the only groups of order 2 are i,j,k,1 and this does not form a subgroup. Reference- http://en.wikipedia.org/wiki/Quaternion_group Message me if you have any doubts

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