Isomorphisms, Products and Decomposition

Isomorphisms

Definition: Let $G$ and $H$ be groups. An isomorphism $f:G\to H$ is a bijective map with the property that, for all $a$,$b$ in $G$, we have $f(ab)=f(a)f(b)$. If $G$ and $H$ are two groups and there exists an isomorphism $f:G\to H$, we say that $G$ and $H$ are isomorphic.

Examples of isomorphic and non-isomorphic groups

The additive group of reals $\mathbb{R}$ and the multiplicative group of positive reals $\mathbb{R}_{+}^{\star}$ are isomorphic. One isomorphism $f:\mathbb{R}\to\mathbb{R}_{+}^{\star}$ is the exponential map $f(x)=e^{x}$.
$\mathbb{Z}_{4}$ is isomorphic to the subgroup $\langle i \rangle\subset\mathbb{C}^{\star}$ via the map sending $f(x)=i^{x}$.
If $a\in\mathbb{Q}^{\star}$ and $a>1$, the subset $\langle a \rangle=\{a^{n} : n\in\mathbb{Z}\}$ is isomorphic to $\mathbb{Z}$ via the map $f(n)=a^{n}$.
$\mathbb{Z}_{n}$ and $\mathbb{Z}_{m}$ are not isomorphic when $n\not=m$ because they have different numbers of elements.
$\mathbb{Z}_{6}$ and $S_{3}$ both have six elements, but they are not isomorphic because one is abelian and one is not.

Properties of isomorphisms.

See this video and these notes.

Theorem: Let $f:G\to H$ be an isomorphism between $G$ and $H$. Then:

$f^{-1}$ is an isomorphism from $H$ to $G$.
$G$ and $H$ have the same number of elements.
if one of $G$ or $H$ is abelian, so is the other.
if one of $G$ or $H$ is cyclic, so is the other.
if one of $G$ or $H$ has a subgroup of order $n$, so does the other.

Some classification results

See this video and these notes.

Theorem: The isomorphism of groups is an equivalence relation among groups.

Any infinite cyclic group is isomorphic to $\mathbb{Z}$.
Any finite cyclic group of order $n$ is isomorphic to $\mathbb{Z}_{n}$.
Any group of order $p$, where $p$ is prime, is isomorphic to $\mathbb{Z}_{p}$.

Cayley’s Theorem

See this video and these notes.

Theorem: Any group $G$ is isomorphic to a group of permutations. If $G$ is finite, then $G$ is isomorphic to a subgroup of $S_{n}$ for some positive integer $n$.

Direct Products

See this video and these notes.

If $G$ and $H$ are groups, the cartesian product $G\times H$ can be made into a group by componentwise multiplication.

Theorem: The group $\mathbb{Z}_{m}\times\mathbb{Z}_{n}$ is isomorphic to $\mathbb{Z}_{mn}$ if and only if $\mathop{gcd}(m,n)=1$.

Corollary: (Decomposition theorem for $\mathbb{Z}_{n}$) Suppose that $n=p_{1}^{e_{1}}\cdots p_{k}^{e_{k}}$ is the prime factorization of $n$. Then $\mathbb{Z}_{n} = \mathbb{Z}_{p_{1}^{e_{1}}}\times\cdots\times\mathbb{Z}_{p_{k}^{e_{k}}}.$

Internal direct products

See this video and these notes.

Definition: Let $G$ be a group and let $H$ and $K$ be two subgroups such that $H\cap K={e}$, $hk=kh$ for all $k\in K$ and $h\in H$; and every element $g\in G$ can be written $hk$ for some $h\in H$ and some $k\in K$. Then $G$ is called the internal direct product of $H$ and $K$.

If $G$ is the internal direct product of $H$ and $K$ then $G$ is isomorphic to $H\times K$.

More generally, suppose that $H_1,\ldots, H_n$ are subgroups of a group $G$ which:

generate $G$;
are such that the only element in common between any two $H_{i}$ is the identity;
mutually commute (so that $h_ih_j=h_jh_i$ for any $h_i\in H_{i}$ and $h_j\in H_{j}$)

Then $G$ is the internal direct product of the $H_{i}$ and is isomorphic to their direct product.

Examples of internal direct products

The dihedral group of the hexagon $D_{6}$ is the internal direct product of two subgroups $H$ and $K$, where $H=\{e, r^{3}\}$ and $K = \{e, r^{2}, r^{4}, s, r^{2}s, r^{4}s\}.$
$S_{3}$ is not an internal direct product of any two of its non-trivial subgroups.