Group

A group is a Monoid with the additional requirement that every element have an inverse. A group need not be commutative; this is a special case known as an Abelian group. #m/def/group

Inverse for each $𝑎 \in 𝐺$ there exists (provably unique) $𝑎 - 1 \in 𝐺$ such that $𝑎 - 1 𝑎 = 𝑒 = 𝑎 𝑎 - 1$

Groups play an important role in describing Symmetry. The concept of a group may be generalised to the concept of a Groupoid, which can be thought of as a typed group.

Terminology and notation

Typically, given a group $𝐺$ the identity element is denoted $𝑒$ (for Einheit). Usually multiplicative notation is used so that juxtaposition or $\cdot$ is the group operation and $𝑎 𝑛$ represents repeated operation. For some abelian groups addition notation may be used where $+$ is the group operation and $𝑛 𝑎$ represents repeated operation.

Both groups and group elements can be assigned order.
A subset of a group that remains closed under the operation is a Subgroup. ${𝑒}$ is the trivial subgroup.

Examples

See Examples of groups

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