# Determine whether or not the two given groups are isomorphic.

1. $(\mathbb{Z},+)$ and $(\mathbb{Z}, *)$ where $a*b=a+b-1$
2. $G$ and $G\times G$, where $G=\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2 \cdots$, one copy of $\mathbb{Z}_2$ for each positive integer.
3. $(\mathbb{Q},+)\times (\mathbb{Q},+)$ and $(\mathbb{Q},+)\times (\mathbb{Z},+)$

Can any one help me with these three? Either hint or answer will be greatly appreciated. I just don't know what group property to look for to see whether they are isomorphic or not. Or in the first place I don't even have an intuition about whether they are isomorphic, and using a specific mapping doesn't really help. For this reason I don't even know whether I should try to prove that they are isomorphic, or I should look for some unsatisfied group property. Any comments or critics on my approach are also appreciated.

Thanks everyone.

1. Work out what the identity and inverses are in $\langle\Bbb Z,\ast\rangle$ to see just what its structure is, and try to compare it with $\langle\Bbb Z,+\rangle$.
2. There is a bijection between $\Bbb Z$ and $\Bbb Z\times\Bbb Z$.
3. Is it possible to solve the equation $x+x=a$ for every $a$ in the first group? What about the second?
• @learnmore: No, the second group is cyclic. What do you have for the identity, and if $n\in\Bbb Z$, what is the inverse of $n$? – Brian M. Scott Oct 29 '14 at 4:10
• My identity is 1 & inverse of any $a$ is $2-a$ – Learnmore Oct 29 '14 at 4:11
• @learnmore: Good; those are right. Now what happens when you calculate $2,2*2,2*2*2,2*2*2*2,\ldots$? You get ... ? – Brian M. Scott Oct 29 '14 at 4:13