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If X is a nonempty subset of a group G,
let <X>={x1^(k1),x2^(k2)...xm^(km)|m>=1, xiEX and kiEZ for each i}.
a) show that <X> is a subgroup of G that contains x.
b) show that <X>C=H for every subgroup H such that XC=H. Thus <X> is the smallest subgroup of G that contains X, and is called the subgroup generated by X.

note: E denotes element of, and C= denotes set containment

Solution Preview

(a) Let A,B are two elements in <X>, then we can express A and B as
A=x1^(k1)*x2^(k2)*...*xm^(km), B=x1^(t1)*x2^(t2)*...*xm^(tm). Thus we have ...

Solution Summary

A proof involving subgroups is provided. The proof is concise.