Subgroup (Mathematics)

up:: Group (Mathematics)

Given a group $(G,\cdot )$ and a subset $H\subset G$, then $(H,\cdot )$ is a subgroup of $G$ if it is a group in its own right under the restricted operation $\cdot {\mid }_H:H\times H\to G$. That is, $H$ is closed under the group operation. Thus, the following properties must hold:

1. Closed under operation: $\forall h,h^{\prime }\in h,h\cdot h^{\prime }\in H$
2. Existence of neutral element: $\exists e_H\in H\mid \forall h\in H,h\cdot e_H=e_H\cdot h=h$
3. Existence of inverse element: $\forall h\in H,\exists h^{-1}\in H\mid h\cdot h^{-1}=h^{-1}\cdot h=e_H$

One denotes $H\le G$ as a shorthand for $H$ being a subgroup of $G$.

Corollaries

The subgroup test can be simplified as

$$\forall h,k\in H,h{k}^{-1}\in H$$

1. $e_H=e_G$, since $h\in H⟹e_H=h\cdot h^{-1}=e_G$

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References

• Subgroup - Wikipedia