(2) Let G be a group of order n, a be an element of G (arbitrary element). Prove that thereexists a positive integer m with 0 < m . Then you can try to argue |H| > |G|, whichthegroupis absurd.)

Subgroup generated by any element in a finite group will have at most n element where n is order of…

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 28E: Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is...

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Question

(2) Let G be a group of order n, a be an element of G (arbitrary element). Prove that there
exists a positive integer m with 0 < m <n such that am = e. (Hint: If not, write down
H generated by a, i.e H =< a >. Then you can try to argue |H| > |G|, which
the
group
is absurd.)

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