# The Klein 4-group vs. the integers modulo 4

Let $(V,٭)$ be a group where $V=\{a,b,c,d\}$. $(V,٭)$ has the property that every element is an inverse of itself, so $V$ is called the $V$ group or the "Klein 4 group".

I would like to know whether $(V,٭)$ is isomorphic to $(\mathbb{Z}_4,+_4)$. How may I achieve this? Should I try to show that every element of $(\mathbb{Z}_4,+_4)$ is also an inverse of itself? but I think it may not be!

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You can achieve this by noting that not every element of $(Z,+_4)$ is the inverse of itself, and concluding that the two groups cannot be isomorphic. – Arturo Magidin Dec 5 '11 at 3:51
@ Arturo;thank u,am okay now – neema Dec 5 '11 at 4:14
It's bad for questions to remain unanswered. How about posting a proof that $V$ is not isomorphic to $\mathbb{Z}_4$ as an answer to your own question? – Arturo Magidin Dec 5 '11 at 4:15
@neema You can prove that the only two groups of order 4 up to isomorphism are the Klein 4-group and the cyclic group of order 4. Since $(\mathbb{Z}_4, +_4)$ is isomorphic to the cyclic group of order 4 it cannot be isomorphic to the Klein 4-group. – user38268 Dec 5 '11 at 5:32

Firstly, we might consider the inverses of each element. In the Klein group, every element is its own inverse. In $\mathbb{Z}_4$, neither $1$ ($1 + 1 = 2$) nor $3$ ($3 + 3 = 2$) are their own inverses while $0$ and $2$ are. So they're not isomorphic.
Secondly, we might consider the subgroups of each. What are the subgroups of $\mathbb{Z}_4$? There are only 3 subgroups: the two trivial subgroups and the group $\langle 0, 2 \rangle$. (If $1$ is in the subgroup, then so are $1 + 1 = 2, 1 + 1 + 1= 3$, and $1 + 1 + 1 + 1 = 0$; if 3 is in the subgroup, then so are $3 + 3 = 2, 3 + 3 + 3 = 1, and 3 + 3 + 3 + 3 = 0$). What about the Klein group? There are at least 5 different subgroups: the two trivial groups, and the groups $\langle a,b \rangle$, $\langle a,c\rangle$, and $\langle a,d\rangle$ (where I assumed that $a$ was the identity element). So they're not isomorphic.
Another way equivalent to the second (and let's be honest - to the first as well) is to consider the orders of the elements. In the Klein group, there are 3 elements of order 2 and the identity. In $\mathbb{Z}_4$, there are 2 elements of order 4 ($1$ and $3$), an element of order 2 ($2$), and the identity. So they're not isomorphic.