- #1

mathusers

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(i) Use sylow's theorem to show that G has a normal subgroup K with [itex]K \cong G [/itex]

(ii) Use the Recogition Criterion to show [itex] G \cong C_p \rtimes_h C_q [/itex] for some homomorphism [itex] h:C_q \rightarrow Aut(C_p) [/itex]

(iii) Describle explicitly all homomorphisms [itex] h:C_5 \rightarrow Aut(C_7) [/itex]. Hence describe all groups of order 35. How many such subgroups are there?

(iv) Describe explicitly all homomorphisms [itex] h:C_3 \rightarrow Aut(C_{13}) [/itex]. Hence describe all groups of order 39. How many such groups are there, up to isomorphism?

any help is highly appreciated as usual. i will attempt the rest myself once i have good idea. thnx a lot :)