Subgroup

algebra

Definition

Subgroup

A group $(U,\circ )$ is a subgroup of group $(G,\circ )$, denoted as $U\le G$, if the following axioms are fulfilled:

1. Non-Empty Subset: $U\subseteq G\wedge U\ne \varnothing$
2. Associativity: $\forall a,b,c\in U:a\circ (b\circ c)=(a\circ b)\circ c$
3. Closure: $\forall a,b\in U:a\circ b\in U$
4. Neutral Element: $\exists e\in U:\forall x\in U:x\circ e=e\circ x=x$
5. Inverse Element: $\forall a\in U:\exists a^{\prime }\in U:a\circ a^{\prime }=a^{\prime }\circ a=e$

Example:

• $(\mathbb{Q},\cdot )\le (\mathbb{R}\setminus \{0\},\cdot )$
• $(m\mathbb{Z},+)\le (\mathbb{Z},+)$
• $(\mathbb{N},+)\cancel{\le }(\mathbb{Z},+$ (not inversible)

Subgroup Criteria

Subgroup Criteria

\begin{align} U \leq G &\Longleftrightarrow \begin{cases} 1. & U \neq \emptyset \\ 2. & \forall x, y \in U: x \circ y \in U \\ 3. & \forall x \in U: x^{-1} \in U

\end{cases} \ \

&\Longleftrightarrow \begin{cases}

1. & U \neq \emptyset \
2. & \forall x, y \in U: x \circ y^{-1} \in U \end{cases} \end{align}