Exercise 3 Let G be a group of order 52-7²-19 = 23275. (a) Prove that G contains exactly one subgroup of order 49. Prove

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Exercise 3 Let G Be A Group Of Order 52 7 19 23275 A Prove That G Contains Exactly One Subgroup Of Order 49 Prove 1 (22.28 KiB) Viewed 69 times

Exercise 3 Let G be a group of order 52-7²-19 = 23275. (a) Prove that G contains exactly one subgroup of order 49. Prove furthermore that if N <G with |N| = 49 then N is normal. (b) Prove that G/N is isomorphic to either Z₁9 x Z25 or Z19 XZs x Zs Suggestion: Similar to the Exercise 2c, exept apply Proposition 3.7.1 instead of 3.7.7. (e) Let Ps and Pis be Sylow 5- and 19-subgroups of G, respectively. Prove that NP, and NP19 are both subgroups of G and that NPN X Ps, and NP19 NX P19- Hint: Corollary 3.6.10 and Proposition 3.7.1.