Algebraic groups

Overview and basic definition

Definition: Group

A group is a set $G$ equipped with a map $\circ :G\times G\to G$, called multiplication or product, which satisfies the following:

• (i) Associativity: $(g\circ h)\circ k=g\circ (h\circ k)$ for all $g,h,k\in G$;
• (ii) Identity: there exists an identity element $e\in G$ such that $g\circ e=e\circ g$ for any $g\in G$;
• (iii) Inverses: for all $g\in G$, there exists $g^{-1}\in G$ such that $g\circ g^{-1}=e$.

If $G$ has finitely many elements, we call $G$ the order of the group $G$. We say $G$ is trivial in the case that $G=\{e\}$.

Relevant theorems:

• (Theorem) Lagrange

Related notes:

• Group homomorphisms and isomorphisms
• Abelian groups
• Normal subgroups and quotient groups

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Subgroups and cosets

Definition: Subgroup

A subgroup of a group $G$ is any subset $H\le G$ which satisfies the following:

• (i) Closure under the group operation: if $a,b\in H$, then $ab\in H$.
• (ii) Inverses and identity: for all $a\in H$, there exists an inverse $a^{-1}\in H$ such that $a{a}^{-1}=e$.

Definition: Left and right cosets, coset space

Given a subgroup $H\le G$, a left $H$-coset of $G$ is the set

$$gH=\{gh|h\in H\}$$

for some $g\in G$. Right cosets $H_g$ are defined similarly. The set $gH$ is sometimes also called a left coset of $H$ with representative $g\in G$.

The left $H$-coset space is the set of al left $H$-cosets of $G$, written

$$G/H=\{gHg\in G\}.$$

The number of left cosets of $H$ in $G$ (which is equivalent to the number of right cosets) is called the index $H$ in $G$ and denoted $|G/H|$.