# $(\{0, 1, … , 2^n-1\}, \oplus)$ groups

I know $(\{0, 1\}, \oplus)$ (where $\oplus$ is the XOR operator) is a well-known abelian group.

But what if i "extend" the underlying set?

For example, i can get $(\{0, 1, 2, 3\}, \oplus)$, or in general $(\{0, 1, ... , 2^n-1\}, \oplus)$ and they are groups as well.

Have these groups been studied? Have they any application in CS?

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If I understand you correctly, your groups should be isomorphic to an $n$-fold product of $C_2$, where $C_2$ is the cyclic group with 2 elements (the one you mentioned in the first sentence of your answer). I don't know how much there is to study about those groups, as they don't look too mysterious to me. I don't know about applications in CS either... –  Sebastian Jan 30 '11 at 13:28
Also, they are just the vector spaces over $\mathbb{Z}/2\mathbb{Z}$. –  George Lowther Jan 30 '11 at 13:43
Since $a\oplus a = 0$ for all a, the group consists of involutions only. The only such groups are indeed the ones sebastian suggests, the elementary abelian groups of order $2^n$. (This follows from the fact that $x\mapsto x^{-1}$ is an automorphism and the structure of (finite) abelian groups.) –  Myself Jan 30 '11 at 14:22
The magic words are "elementary abelian $2$-groups". They show up all over the place. –  Arturo Magidin Jan 30 '11 at 19:19

As Sebastian mentioned, the groups are not mysterious at all. In fact, they are quite "boring" since they're Abelian.

Here are some example applications, most of them practical, and some theoretical.

### CPUs

Your CPU supports an operation called XOR (Exclusive OR) which views the registers as members of the group $\mathbb{Z}_2^n$ (where $n=32$ or $n=64$ for modern CPUs), and applies the operation of the group. The CPU also supports addition in the group $\mathbb{Z}_{2^n}$, i.e. addition modulo $2^n$, as well as multiplication modulo $2^n$.

### Coding Theory

Lots of codes are defined using these groups. CRCs are linear, so they can be viewed as a linear mapping from $\mathbb{Z}_2^n$ to $\mathbb{Z}_2^m$ (where $m=16$ or $m=32$). All error-correcting codes used in practice are also linear.