group

Definition

A group is a set $G$ with binary operation (i.e. a law of composition) $*$ satisfying axioms of:

• closure
  • for all elements $g$ and $h$ of $G$, $g * h$ is an element of $G$
• associativity
  • $(g * h) * k = g * (h * k)$ for all $g,h,k \in G$
• existence of identity
  • exists element $e \in G$, identity (or unit) of $G$ such that $e * g = g * e = g$ for all $g \in G$
• existence of inverse
  • for every $g \in G$ there exists an element $h \in G$ called the inverse of $g$ such that $e * g = g * e = g$ for all $g \in G$

An abelian or commutative group is a group $(G,*)$ also satisfying

• commutativity
  • $g * h = h * g$ for all $g,h \in G$

Notes

• a group is a special case of semigroup where there is existence of identity and inverse
• it can also be considered as a special case of a monoid with an inverse
• closure is sometimes omitted in definitions, in this case, it is incorporated into the properties required for the relevant law of composition instead

References

1. https://people.tamu.edu/~yvorobets/MATH433-2010B/Lect2-05web.pdf
2. https://www.bananaspace.org/wiki/群
3. https://zhuanlan.zhihu.com/p/314567658
4. https://math.stackexchange.com/questions/63697/why-is-closure-omitted-in-some-group-definitions
5. M. Artin, Algebra, 2. ed. Boston, Mass. Munich: Pearson Education, Prentice Hall, 2011, pp. 42-43.