Coset containment implies subgroup containment

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Statement

Verbal statement

If a Left coset (?) of one subgroup is contained in a left coset of another subgroup, then the subgroup is also contained in the other.

Statement with symbols

Suppose xH \subseteq yK are cosets of subgroups H,K \le G. Then H \le K.

Related facts

Proof

Given: xH \subseteq yK for some x,y \in G.

To prove: H \le K

Proof: Since x=xe is in xH, which is contained in yK, there exists k_1 \in K such that x=yk_1. Thus yk_1H \subseteq yK, which implies that there exist h \in H and k_2 \in K such that yk_1h=yk_2. Hence h=k_2k_1^{-1}, which is in K. So, H\le K.