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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?"


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They are Boolean groups. – Brian M. Scott Jan 9 '13 at 17:12
@BrianM.Scott- Thanks, that's perfect! Can you post that as an answer so I can accept it? – templatetypedef Jan 9 '13 at 17:13
Or $\Bbb Z_2\times\mathbb Z_2$ – Babak S. Jan 9 '13 at 17:14
Is $3$ an inverse of itself under addition modulo $4$? – Ilya Jan 9 '13 at 17:15
@Ilya- Oh whoops, you're right. Let me fix that... – templatetypedef Jan 9 '13 at 17:16
up vote 8 down vote accepted

They are often called Boolean groups.

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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).

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

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