Subgroup

A subgroup $(H,\times_H:H\times H\rightarrow H$ of a Group $(G,\times_G:G\times G\rightarrow G)$ is a set $H\subseteq G$ which is a group under the operation $\times_G$ restricted to $H\times H$.

Definition

Given a group $(G,\times_G:G\times G\rightarrow G)$ we say $(H,\times_H:H\times H\rightarrow H)$ is a subgroup of $(G,\times_G)$ if:

1. $$H\subset G$$
2. the function $$\times_H:H\times H\rightarrow G$$ given by $$\times_H(x,y)\mapsto\times_G(x,y)$$ has $$\text{Range}(\times_H)\subseteq H$$
   • That is to say it is closed. $$\forall x\in H\forall y\in H[\times_H(x,y)\in H]$$
3. There exists an identity element $$\in H$$ .
   • That is to say $$\exists e\in H\forall x\in H[ex=xe=x]$$ where $$xy$$ denotes $$\times_H(x,y)$$
4. Every element has an inverse $$\in H$$
   • That is to say $$\forall x\in H\exists y\in H[xy=yx=e]$$
5. The operation is associative
   • That is to say $$\forall x\in H\forall y\in H\forall z\in H[x(yz)=(xy)z]$$

Just like a group

This makes it sound a lot harder than it really is.

Examples

Even numbers

Take the group

$$(\mathbb{Z},+)$$ and define $$H=\{z\in\mathbb{Z}|z\text{ is even}\}$$ then we have $$H\subset\mathbb{Z}$$ and we must check it is a group.

1. It is closed under $$+$$ restricted to $$H$$ - an even + an even = even. (proof $$2n+2m=2(m+n)$$ and anything multiplied by 2 is even)
2. The identity $0\in H$ - so we have that.
3. Given an $$x\in H$$ we can see easily that the inverse, $$-x$$ is also even and thus $$\in H$$
4. Associativity is inherited

See also

• Coset
• Normal subgroup