Let G be a group of order 60. If the Sylow 3-subgroup is normal,show that the Sylow 5-subgroup is normal.

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Q: Let H and K be normal subgroups of a group G such at HCK, show that K/H is a normal subgroup of G/H.

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Q: Let G be a group of order 2p, where p is  prime. Show that G has a normal subgroup  of order p.

A: G be a group of order 2p, where p is prime.    Claim : G has a normal subgroup of order p.

Q: Show that if a Sylow P subgroup is normal then it is characteristic

A: If a Sylow subgroup is normal, then it is the unique subgroup of that order, i.e. if P◃G

Q: Find all subgroups of Z60 and draw a lattice diagram for them.

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Q: 8. Show that if |G| = 12 then G has a normal Sylow 2-subgroup or a normal Sylow 3-subgroup.

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Q: (4) Let G be a group. Prove that Z(G) is a normal subgroup of G.

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Q: Answer the followings: 1. Let H = {[a b]: a, b, d € R, ad # 0}. Is H a normal subgroup of GL₂(R)? 2.…

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