Term for a group where every element is its own inverse?

Question

Several groups have the property that every element is its own inverse. For example, the numbers $0$ and $1$ and the XOR operator form a group of this sort, and more generally the set of all bitstrings of length $n$ and XOR form a group with this property.

These groups have the interesting property that they have to be commutative.

Is there a special name associated with groups with this property? Or are they just "abelian groups where every element has order two?"

Thanks!

score 12 · Accepted Answer · answered Jan 09 '13 at 17:15

12

They are often called Boolean groups.

answered Jan 09 '13 at 17:15

Brian M. Scott

616,228

score 9 · Answer 2 · answered Jan 09 '13 at 17:19

9

Another term for these groups is elementary abelian $2$-groups. In general, an elementary abelian $p$-group (for a prime $p$) is an abelian group where every non-identity element has order $p$ (and it is easy to see that if all non-identity elements have the same order, then that order must be a prime).

answered Jan 09 '13 at 17:19

Tobias Kildetoft

21,247

Be careful though, because the word "2-group" can have a completely different meaning... – silvascientist Oct 02 '18 at 18:56
1

@MonstrousMoonshiner I would rather say that anyone wanting to use the term for the higher categorical stuff needs to make sure not to cause confusion, seeing as they could easily have chosen so many other names for it (or at least added an "oid" or something). – Tobias Kildetoft Oct 02 '18 at 19:06

score 5 · Answer 3 · answered Jan 09 '13 at 17:17

5

These are (the underlying additive groups of) the vector spaces over $\Bbb{Z}/2\Bbb{Z}$.

answered Jan 09 '13 at 17:17

Chris Eagle

33,306

score 0 · Answer 4 · answered Oct 02 '18 at 18:41

0

Another term is "group of exponent $2$".

answered Oct 02 '18 at 18:41

lhf

216,483

Term for a group where every element is its own inverse?

4 Answers4