A Grand Truth

Group theory is a branch of math­e­mat­ics that stud­ies sym­me­try and struc­ture through sets equipped with a spe­cial op­er­a­tion. It is es­sen­tial not only in pure math­e­mat­ics such as number theory, but also in physics, chem­istry, and com­put­er sci­ence.

A group is a set $G$ to­geth­er with a binary op­er­a­tion $\star$, which com­bines any two el­e­ments to form an­oth­er el­e­ment of the set, and sat­is­fies four fun­da­men­tal prop­er­ties:

• Clo­sure: For all $a, b \in G$, the result $a \star b$ is also in $G$.
• As­so­cia­tiv­i­ty: For all $a, b, c \in G$, the group­ing of op­er­a­tions does not matter:

Here, $a$, $b$, and $c$ are el­e­ments of $G$.

• Iden­ti­ty El­e­ment: There is an el­e­ment $e \in G$, called the iden­ti­ty (pro­nounced "ee-dent-uh-tee"), such that for every $a \in G$:

The iden­ti­ty el­e­ment leaves any el­e­ment un­changed under the group op­er­a­tion.

• In­verse El­e­ment: For each el­e­ment $a \in G$, there exists an in­verse el­e­ment $a^{-1} \in G$ (read as "a in­verse"), such that:

The in­verse el­e­ment "undoes" the effect of $a$ under the group op­er­a­tion.