^^ [[Schutz - 1 - Some Basic Mathematics|1. Some Basic Mathematics]] << [[Schutz - 1.3 - Real Analysis|1.3 Real Analysis]] | [[Schutz - 1.5 - Linear Algebra|1.5 Linear Algebra]] >> # Group Theory > **Group** > A collection of elements $G$ together with a binary operation $\bullet$ is called a *group* if it satisfies the axioms: > 1. *Associativity*: For $x, y, z \in G, \quad x \bullet (y \bullet z) = (x \bullet y) \bullet z$ > 2. *Right identity:* $G$ contains an element $e$ such that, for any $x$ in $G$, $x \bullet e = x$ > 3. *Right inverse:* for every $x$ in $G$ there is an element called $x^{-1}$, also in $G$, for which $x \bullet x^{-1} = e$ > A group is *Abelian* (or *commutative*) if it additionally satisfies: >4. *Commutativity*: $x\bullet y = y \bullet x \, \forall x, y \in G$ ^426790 Note that from (1-3) we can conclude that $e$ is unique, and that it is also a left-identity (i.e. $e \bullet x = x$). The inverse element $x^{-1}$ is also unique for any $x$, and it is also a left inverse (i.e. $x{-1} \bullet x = e$). > **Continuous Group** > A [[Schutz - 1.4 - Group Theory#^426790|group]] is a *continuous group*, if there is a bijection from any open set of elements of the group to an open set of $\mathbb{R}^n$ for some $n$. > **Subgroup** > A *subgroup* $S$ of a group $G$ is a collection of elements of $G$ which themselves form a group with the same binary operation. As a group, the subgroup must contain an identity element. Since $e \in G$ is unique, we can conclude that $e \in S$. > **Group Isomorphism** > Consider two groups $G_1$ and $G_2$ with binary operations $\bullet$ and $*$ respectively. They are called *isomorphic* (i.e. identical in their group properties) if there is an [[Schutz - 1.2 - Mappings#^6c8b58|bijection]] from $G_1$ to $G_2$ which respects the group operations. > For $x, y \in G_1$ and $f:G_1 \rightarrow G_2$, then > $ f(x \bullet y) = f(x) * f(y) $ > We call $f$ a *group isomorphism*. ^fc423a > **Group Homomorphism** > A *group homomorphism* is like an [[Schutz - 1.4 - Group Theory#^fc423a|isomorphism]] except it can be *many-to-one* and may only be *into*. Like above, we still have, for $x, y \in G_1$ and $f:G_1 \rightarrow G_2$, > $ f(x \bullet y) = f(x) * f(y) $