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

then N is normal.

(b) Prove that G/N is isomorphic to either Z19 × Z25 or Z19 × Z5 X Z

Suggestion: Similar to the Exercise 2c, exept apply Proposition 3.7.1 instead of 3.7.7.

49

(c) Let Ps and P19 be Sylow 5- and 19-subgroups of G, respectively. Prove that NP5 and NP19 are both subgroups

of G and that

NPN X P5, and NP19 N x P19-