32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same proof shows that the intersection of any number of subgroups of G, finite or infinite, is again a subgroup of G?)

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Q: Suppose H and K are subgroups of a group G. If |H| = 12 and|K| = 35, find |H ⋂ K|. Generalize.

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Q: Show that S = SU(2) contains a subgroup isomorphic to S'.

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Q: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

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Q: 9. Prove that H ne Z} is a cyclic subgroup of GL2(R). . Subgraup chésed in Pg 34

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Q: 37. Is the set Ze of even integers a subgroup of Z with addition? What's about (Z,,+)?

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Q: 1) If (H, *) is a subgroups of (G, *)then (NG(H) , * ) is a subgroup of (G, *).

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Q: Is every subgroup of Z cyclic? Why? Describe all the subgroups of Z.

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Q: 5. If H. aEA are a family of subgroups of the group G, show that is a subgroup of G.

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Q: 7. Prove that if G is a group of order 1045 and H€ Syl₁9 (G), K € Syl (G), then KG and HC Z(G).

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Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups that G has is 4 O 12 O 18…

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A: Need to find intersection of subgroup