3
$\begingroup$

suppose you have two sets $G_1$ and $G_2$ with same cardinality

in $G_2$ you have the group structure and there is a bijective map from $G_1$ to $G_2$ this is just a set map. can we define a binary operation on $G_1$ with the help of binary operation on $G_2$ so that $G_1$ also become a group?

  • 0
    The magic words are [transport of structure](http://en.wikipedia.org/wiki/Transport_of_structure).2012-05-20

1 Answers 1

6

HINT: There’s really only one reasonable thing to try, and it works. Let $f:G_1\to G_2$ be your bijection, and let $\cdot$ be the group operation in $G_2$. You want to use $f$ to define a group operation $\odot$ on $G_1$. Suppose that $x,y\in G_1$; what would $x\odot y$ have to be in order for $f$ to be a group isomorphism?