Find out if the following pairs of groups are isomorphic:
1 $(\mathbb {Z}_{7}\setminus\{0\}; .)$ with $(\mathbb {Z}_{6}; +)$
2 $(\mathbb {R}_{+}; .)$ with $(\mathbb {R}\setminus\{0\}; .)$
3 $(\mathbb {R}\setminus\{0\}; .)$ with $(\mathbb {C}\setminus\{0\}; .)$
4 $(\mathbb {R}; +)$×$(\mathbb {R}; +)$ with $(\mathbb {C}; +)$
5 $(\mathbb {Z}_{2}; +)$×$(\mathbb {Z}_{2}; +)$ with $(\mathbb {Z}_{4}; +)$
6 the group of symmetries of the equilateral triangle with group of the all permutations of {$1,2,3$}.
I know that I have to find some bijective function which is homomorphism. But it takes very long time to confirme all possibilities. I would like to now if there is any general way how to find out if two groups are isomorphic.
Thank you very much for any help.
The groups from item 1 are isomorphic because they are both cyclic groups of order $6$ ($(\mathbb{Z}_7\setminus\{0\},.)$ is generated by $3$).
The groups of item 2 are not isomorphic, because $(\mathbb{R}_+,.)$ has no element of order $2$, whereas $(\mathbb{R}\setminus\{0\},.)$ has such an element ($-1$).
In general, examining the orders of the elements is a good strategy to prove that two groups are not isomorphic.
Can you take it from here?