Exercise 2 Let G be a group of order 3².52.13=2925. (a) Prove that G contains exactly one subgroup of order 25 and exact

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Exercise 2 Let G Be A Group Of Order 3 52 13 2925 A Prove That G Contains Exactly One Subgroup Of Order 25 And Exact 1 (32.76 KiB) Viewed 73 times

Exercise 2 Let G be a group of order 3².52.13=2925. (a) Prove that G contains exactly one subgroup of order 25 and exactly one subgroup of order 13. Hint: Use the Sylow theorems. (b) Let N and K with be subgroups of G with |N|=25 and |K|=13 (proven to exist in (s)). Prove that both N and K are normal. (e) Prove that G/N is isomorphic to either Z₁3 x Zo or Z₁3 x (Z₁ x Z). Suggestion: Let G = G/N and apply the 3rd Sylow theorem to G. Next, apply Ptoposition 3.7.7 to G. Finally, apply Exercise 4.3.4 to the Sylow 3-subgroup of G to determine the structure of the Sylow 3-subgroup of G.