Groups Review

Groups Review

We will now review some of the recent material regarding our introduction to groups.

• On the Groups page we formally defined an algebraic structure known as a group. Recall that if $*$ is a binary operation on $G$ then $G$ is a Group under the operation $*$ denoted $(G, *)$ if the following properties are satisfied:
• For all $a, b \in G$ we have that $a * b \in G$ (Closure under $*$).
• For all $a, b, c \in G$ we have that $a * (b * c) = (a * b) * c$ (Associativity of elements in $G$ under $*$).
• There exists an element $e \in G$ such that $a * e = a$ and $e * a = a$ (Existence of an identity element in $G$ for $*$).
• For all $a \in G$ there exists an $a^{-1} \in G$ such that $a * a^{-1} = e$ and $a^{-1} * a = e$ (Existence of inverses elements for each element in $G$ under $*$).
• Also recall that $(G, *)$ is a Finite Group if $G$ is a finite set and is an Infinite Group if $G$ is an infinite set. The Order of $(G, *)$ is defined to be the number of elements in $G$ and is denoted $\mid G \mid$.
• On the Subgroups and Group Extensions page we said that if $(G, *)$ is a group, $H \subseteq G$, and $(H, *)$ forms a group then $(H, *)$ is said to be a Subgroup of $(G, *)$. Furthermore, $(G, *)$ is said to be a Group Extension of $(H, *)$.
• We then looked at a very nice theorem which gives us a simpler criterion for determining whether a subset of a group is a subgroup. We proved that $(S, *)$ is a subgroup of $(G, *)$ if and only:
• $S$ is closed under $*$.
• For every $a \in S$ there exists an $a^{-1} \in S$ such that $a * a^{-1} = e$ and $a^{-1} * a = e$.
• On the Basic Theorems Regarding Groups page we looked at quite a few elementary results regarding groups which are summarized in the following table. Let $(G, *)$ be a group and let $e \in G$ be the identity element.
Theorem
The identity element $e$ is unique.
For every $a \in G$, $a^{-1}$ is unique.
For every $a \in G$, $(a^{-1})^{-1} = a$.
For every $a, b \in G$, $(a * b)^{-1} = b^{-1} * a^{-1}$.
For every $a, b \in G$, if $a * b = e$ then $a = b^{-1}$ and $b = a^{-1}$.
For every $a \in G$, if $a^2 = a * a = a$ then $a = e$.
• On the The Cancellation Law for Groups page we noted a very important result regarding groups called the Cancellation Law which says that if $(G, *)$ is a group and $a, b, c \in G$ are such that $a * b = a * c$ or $b * a = c * a$ then $b = c$.
(1)
\begin{align} \quad x^n = \underbrace{x * x * ... * x}_{n \: \mathrm{times}} \end{align}
• Furthermore, if $x^n = a$ then $x$ is said to be an $n^{\mathrm{th}}$ Root of $a$.
• On the The Intersection and Union of Two Subgroups page we looked at two results regarding groups. We saw that if $(G, *)$ is a group and $(S, *)$, $(T, *)$ are two subgroup of $(G, *)$ then we ALWAYS have that $(S \cap T, *)$ is also a subgroup of $(G, *)$.
• However we proved that $(S \cup T, *)$ is a subgroup of $(G, *)$ if and only if $S \subseteq T$ or $S \supseteq T$.
• We then looked at a bunch of examples of groups which are linked below: