Groups, Order and Subgroups

Groups

Group

A group is a set $G$ together with a composition law $\circ :G\times G\to G$ such that the following properties hold:

• Associativity: $(a\circ b)\circ c=a\circ (b\circ c)$ for all $a,b,c\in G$.
• Identity: there exists $e\in G$ such that $e\circ a=a\circ e=a$ for all $a\in G$.
• Inverse: for every $a\in G$ there exists $b\in G$ such that $a\circ b=b\circ a=e$.

We may sometime denote the group by $(G,\circ )$ to emphasize the law of composition.

Proposition

Let $G$ be a group. Then

1. For any $a\in G$, the inverse is unique and denoted $a^{-1}$.
2. For any $a\in G$, $(a^{-1}{)}^{-1}=a$.
3. $(ab{)}^{-1}=b^{-1}{a}^{-1}$.
4. In $G$ the cancellation laws holds: for $a,b,c\in G$, $ab=ac$ implies that $b=c$.

Subgroups and Order

Generalized Associativity

Whilst the definition of associativity involves only three elements it implies the generalized associative law, that is, a product of $n$ elements in a group $a_1{a}_2\cdot \cdot \cdot a_n$ has the same value regardless of any parentheses.

Abelian Group

A group is called abelian if the composition law is commutative.

Finite Group

A group is called finite if $G$ is a finite set; otherwise the group is called infinite. The order of a finite group is the number of elements in a finite group, denoted $|G|$.

Power

• For an element $a$ and in a group $G$ and $n\in \mathbb{N}$ we write $a^n$ for the unambiguous composition of $a$ with itself $n$ times (this follows from generalized associativity). We also define $a^0:=e$ and $a^{-n}=(a^{-1}{)}^n$.
• An element $g\in G$ has finite order if there is $n\in \mathbb{N}$ such that $g^n=e$; the minimal such $n$ is the order of $g$ and denoted $o(g)$. Otherwise we say that $g$ has infinite order.

Laws of Exponents

Let $G$ be a group, let $a,b\in G$ , and let $m$ and $n$ be integers. Then

1. If $a$ and $b$ commute, then $(ab{)}^n=a^n{b}^n$.
2. $a^m{a}^n=a^{m+n}$.
3. $(a^n{)}^m=a^{mn}$.

Subgroup

Let $G$ be a group, a nonempty subset $H\subset G$ is called a subgroup of $G$ if it is a group under the same law of composition of $G$. In such case we write $H\le G$.

Remark

A nonempty subset $H\subset G$ is a subgroup if and only if it is closed under composition and under taking inverses. That is $a,b\in H⟹ab\in H$ and $a\in H⟹a^{-1}\in H$. The latter two can be consolidated to $a,b\in H⟹a{b}^{-1}\in H$.