Hi next one, bit confused with this problem: any hints on any of the parts would be greatly appreciated.

QUESTION:

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let G be a group of order 8 and suppose that has ord(y)=4. Put and let

(i) show that and that

(ii) list, a priori, the possibilities for and label them .

(iii) list the possibilities for (HINT: CHECK ORDERS) and label them

(iv) By examining the pairs, in turn show that G is isomorphic to one of .

Deduce that an arbitrary group of order 8 is isomorphic to one of

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For part (i) i understand i have to show G is a normal subgroup of H but im not sure how to show that G is a subgroup (i.e. closure holds, existance of identity element of G in H, etc) and it is normal. i..e For all g in G, gHg^−1 ⊆ N.