Group theory and generalizations

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• ConceptGroup

  A group is a set $G$ with a binary operation $\cdot$ that has these three properties. The operation is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c$ in $G$. There is an identity element $e$ in $G$ such that $e \cdot a = a \cdot e = a$ for all $a$ in $G$. Each element $a$ of $G$ has an inverse $a^{-1}$ in $G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.

• ConceptAbelian group

  An abelian group is a group whose operation is commutative. The integers with addition are the basic example.

• ConceptSubgroup

  A subgroup of a group is a subset that is itself a group under the same operation.

• ConceptGroup homomorphism

  A group homomorphism is a map between groups that preserves the group operation. It is the basic way to compare two groups.

• ConceptKernel of a homomorphism

  The kernel of a group homomorphism is the set of elements that it sends to the identity. It is a normal subgroup, and it measures how far the homomorphism is from being injective.

• ConceptAutomorphism group

  The automorphism group of a group $G$ is the group of all isomorphisms from $G$ to itself, with composition as the operation. It describes the symmetries of the structure of $G$.