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 𝑎 ∈𝐺 there exists (provably unique) 𝑎−1 ∈𝐺 such that 𝑎−1𝑎 =𝑒 =𝑎𝑎−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 ⋅ is the group operation and 𝑎𝑛 represents repeated operation. For some abelian groups addition notation may be used where + is the group operation and 𝑛𝑎 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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