# Order of a group given group representation

by bt203   Last Updated September 11, 2019 18:20 PM - source

Suppose that $$G$$ is the group generated by $$a,b$$ satisfying the relations $$ba=a^2b, ab=b^2a$$, then how can I find the cardinality of this group? I've tried to rearrange these two relations by doing some cancellation and substitution, but nothing worked out.

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$$ab=b(ba)=ba^2b$$ so $$a=ba^2$$, hence $$ba=1$$ and $$b=a^{-1}$$.
Putting this into $$ba=a^2b$$ gives us $$1=a$$, so $$G$$ is the trivial group.