# Subgroups

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

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#### Solution Preview

Proof:

(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.

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