Suppose G is a group, H a subgroup of G, and a and b elements of G. If a in bH then b in aH.

Abstract algebra
asked 2020-12-15
Suppose G is a group, H a subgroup of G, and a and b elements of G. If \(\displaystyle{a}\in{b}{H}\) then \(\displaystyle{b}\in{a}{H}\).

Answers (1)

Let \(\displaystyle{x}\in{a}{H}\cap{b}{H}\), then there exist \(\displaystyle{h}_{{1}},{h}_{{2}}\in{H}\)
Such that,
(since \(\displaystyle{a}{H}={H}{\quad\text{if}\quad}{a}\in{H}\))
Now, if \(\displaystyle{a}\in{b}{H}\), using above theorem
Since \(\displaystyle{b}\in{b}{H}\), then \(\displaystyle{a}{H}={b}{H}\Rightarrow{b}\in{a}{H}\)
Hence G be a group, H a subgroup of G, and a and b elements of G. If \(\displaystyle{a}\in{b}{H}\) then \(\displaystyle{b}\in{a}{H}\).

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