- #1

mathusers

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QUESTION:

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let G be a group of order 8 and suppose that [itex]y \epsilon G[/itex] has ord(y)=4. Put [itex]H = [1,y,y^2,y^3][/itex] and let [itex]x \epsilon G-H[/itex]

(i) show that [itex]H \lhd G[/itex] and that [itex]x^2 \epsilon H[/itex]

(ii) list, a priori, the possibilities for [itex]x^2[/itex] and label them [itex]P_1,...,P_n[/itex].

(iii) list the possibilities for [itex]xyx^{-1}[/itex] (HINT: CHECK ORDERS) and label them [itex]Q_1,...,Q_m[/itex]

(iv) By examining the pairs, [itex](P_i,Q_j)[/itex] in turn show that G is isomorphic to one of [itex]C_8, C_4 \times C_2, D_8, Q_8[/itex].

Deduce that an arbitrary group of order 8 is isomorphic to one of [itex]C_2 \times C_2 \times C_2, C_4 \times C_2, C_8, D_8, Q_8[/itex]

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