1. Let p: G → H be a group homomorphism. Show: (c) If K is a subgroup of G, then ) p(K) is a subgroup of H. (d) If K' is a subgroup of H, then o-1(K') is a subgroup of G.

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Q: 2- Let (C\{0},.) be the group of non-zero -complex number and let H = { 1,-1, i,-i} prove that (H,.)…

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Q: If H and K are two subgroups of finite indices in G, then show that H ∩ K is also of finite index in…

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Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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Q: If H and K are subgroups of G of order 75 and 242 respectively, what can you say about H N K?

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Q: (a) Let G be a non-cyclic group of order 121. How many subgroups does G have? Why? (b) Can you…

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Q: Let G and H be groups, and let ø:G-> H be a group homomorphism. For xeG, prove that )x).

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Q: 2. Let H and K be subgroups of the group G. (a) For x, y E G, define x ~ y if x = hyk for some h e H…

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