Group theory MOC

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. group

4. Inverse for each there exists (provably unique) such that

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 the identity element is denoted (for Einheit). Usually multiplicative notation is used so that juxtaposition or is the group operation and represents repeated operation. For some abelian groups addition notation may be used where is the group operation and 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. is the trivial subgroup.

Examples

See Examples of groups

---

develop | en | sembr