Section 6 Group Basics

6.1 Groups

Def: A group is a set $\mathsf{G}$ with a binary operation $\ast$ that satisfies the following:

(Identity) There exists $e \in \mathsf{G}$ such that: if $g \in \mathsf{G}$, then $e \ast g= g \ast e = g$.
(Inverse) If $g \in \mathsf{G}$, then there exists $g^{-1} \in \mathsf{G}$ so that $g \ast g^{-1} = g^{-1} \ast g = e$.
(Associativity) If $a,b,c \in \mathsf{G}$, then $(ab)c = a (bc)$.

Furthermore, the group is abelian if it also satisfies the commutative property:

(Commutative) If $a,b \in \mathsf{G}$, then $ab = ba$.

Proposition (Uniqueness) Let $\mathsf{G}$ be a group. Then

$\mathsf{G}$ has a unique identity element $e$.
If $a,b,c \in \mathsf{G}$ and $ab = e$ and $ca = e$, then $a = c$.
If $g \in \mathsf{G}$ then t here is a unique element $g^{-1}$ such that $g g^{-1} = g^{-1} g = e$.

Proposition (Cancellation) Let $\mathsf{G}$ be a group and $a,b,c \in \mathsf{G}$.

If $a b = a c$ then $b = c$.
If $b a = c a$ then $b = c$.

Proposition ($ax =b$) Let $\mathsf{G}$ be a group and $a,b \in \mathsf{G}$.

The equation $a x = b$ has a unique solution $x = a^{-1} b$.
The equation $x a = b$ has a unique solution $x = b a^{-1}$.

Proposition (Latin Square) Let $\mathsf{G}$ be a group and $a \in \mathsf{G}$.

The function $\mathsf{G}\to \mathsf{G}$ defined by left multiplication by $a$, $g \mapsto a g$, is one-to-one and onto.
The function $\mathsf{G}\to \mathsf{G}$ defined by right multiplication by $a$, $g \mapsto g a$, is one-to-one and onto.

6.2 Subgroups

Def: A subset $\mathsf{H}\subseteq \mathsf{G}$ of a group $\mathsf{G}$ is a subgroup if $\mathsf{H}$ is a group under the same binary operation that it inherits from $\mathsf{G}$. We write $\mathsf{H}\le \mathsf{G}$ to denote that $\mathsf{H}$ is a subgroup of $\mathsf{G}$ and write $\mathsf{H}< \mathsf{G}$ if $\mathsf{H}\le \mathsf{G}$ and $\mathsf{H}\not= \mathsf{G}$.

Theorem: (Subgroup Test) A nonempty subset $\mathsf{H}\subseteq \mathsf{G}$ is a subgroup if

(identity) $e \in \mathsf{H}$, where $e$ is the identity in $\mathsf{G}$.
(closure) If $h_1, h_2 \in \mathsf{H}$, then $h_1 h_2 \in \mathsf{H}$.
(contains inverses) If $h \in \mathsf{H}$, then $h^{-1} \in \mathsf{H}$.

Note: (a) follows from (b) and (c).

Def: The order of an element $g \in \mathsf{G}$ is the smallest positive integer $m$ such that $g^m = e$, and we write $|g| = m$. If there is no such $m$, then $|g| = \infty$.

Def: The center of a group $\mathsf{G}$ is the following subgroup \[ Z(\mathsf{G}) = \{ a \in \mathsf{G}\mid \hbox{ if $g \in \mathsf{G}$ then $ag = ga$ } \}. \]

Def: If $g \in \mathsf{G}$ then $\langle g \rangle$ is the smallest subgroup of $\mathsf{G}$ that contains $g$. It is called the cyclic subgroup generated by $g$. This subgroup contains all possible powers of $g$: \[ \langle g \rangle = \{ g^k \mid k \in \mathbb{Z}\}. \]

6.3 Cyclic Groups

A group $\mathsf{G}$ is cyclic if $\mathsf{G}= \langle g \rangle$ for some $g \in \mathsf{G}$.

Theorem. (Fundamental Theorem of Cyclic Groups) If $\mathsf{G}= \langle g\rangle$, then

Every subgroup $\mathsf{H}\le \mathsf{G}$ is cyclic (thus, $\mathsf{H}= \langle g^k \rangle$).
If $|g| = \infty$ then $\langle g \rangle = \{ \ldots, g^{-3}, g^{-2}, g^{-1}, 1, g, g^2, g^3, \ldots \}$ and all of these elements are distinct.
If $|g| = n$ then $\langle g \rangle = \{1, g, g^2, \ldots, g^{n-1} \}$ and all of these elements are distinct (thus, $|g| = |\langle g \rangle|$).
If $|g| = n$ then the subgroups of $\langle g \rangle$ satisfy:
1. $\langle g^k \rangle = \langle g^d \rangle$, where $d = \gcd(k,n)$.
2. $|\langle g^k \rangle| = \frac{n}{d}= \frac{n}{\gcd(k,n)}$, which is a divisor of $n$.
3. If $\ell \vert n$, then $|\langle g^{n/\ell} \rangle|$ is a subgroup of size $\ell$ and is the only subgroup of size $\ell$.

Note: (d) says that, in a finite cyclic group of order $n$, there is exactly one subgroup for each divisor of $n$ and those are all the subgroups.

6.4 Isomorphisms

Def: An isomorphism between groups $\mathsf{G}$ and $\mathsf{H}$ is a function $\phi: \mathsf{G}\to \mathsf{H}$ such that $\phi$ is and (a bijection) and \[ \phi( a b) = \phi(a) \phi(b). \] We write $\mathsf{G}\cong \mathsf{H}$ to denote that $\mathsf{G}$ and $\mathsf{H}$ are isomorphic.

Theorem (Properties of Isomorphisms): if $\phi: \mathsf{G}\to \mathsf{H}$ is an isomorphism, and $g then

$\phi(e_\mathsf{G}) = e_\mathsf{H}$.
$\phi(g^{-1}) = \phi(g)^{-1}$
$\phi(g^k) = \phi(g)^k$ for all $k \in \mathbb{Z}$
$|g| = |\phi(g)|$
$\phi^{-1}: \mathsf{H}\to \mathsf{G}$ is an isomorphism
If $K \le \mathsf{G}$ then $\phi(\mathsf{K}) \le H$ where $\phi(\mathsf{K}) = \{ \phi(k) \mid k \in \mathsf{K}\}$.
$|\mathsf{G}| = |\mathsf{H}|$
$\mathsf{G}$ is abelian if and only if $\mathsf{H}$ is abelian
$\mathsf{G}$ is cyclic if and only if $\mathsf{H}$ is cyclic
$\mathsf{G}$ has $k$ elements of order $m$ if and only if $\mathsf{H}$ has $k$ elements of order $m$.