Tour:Subgroup

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Given below is the definition of subgroup. A subgroup of a group is, roughly speaking, a subset that's also a group. The definition of subgroup is given in a number of equivalent ways below. Before proceeding further, make sure that you fully understand why all the definitions are equivalent, and why they tally with whatever definitions you see in textbooks.
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Definition

The universal algebraic definition

Let $G$ be a group. A subset $H$ of $G$ is termed a subgroup if all the three conditions below are satisfied:

• Whenever $a,b$ belong to $H$, so does $ab$ (here $ab$ denotes the product of the two elements)
• Whenever $a$ belongs to $H$, so does $a^{-1}$ (the multiplicative inverse of $a$)
• $e$ belongs to $H$ (where $e$ denotes the identity element)

Definition via the subgroup condition

The equivalence of this definition with the earlier one is often called the subgroup condition. For full proof, refer: Sufficiency of subgroup condition

It has two forms (left and right):

• A subset of a group is termed a subgroup if it is nonempty and is closed under the left quotient of elements. In other words, a subset $H$ of a group $G$ is termed a subgroup if and only if $H$ is nonempty and $a^{-1}b\in H$ whenever $a,b\in H$
• A subset of a group is termed a subgroup if it is nonempty and is closed under the right quotient of elements. In other words, a subset $H$ of a group $G$ is termed a subgroup if and only if $H$ is nonempty and $ab^{-1}\in H$ whenever $a,b\in H$

Definition in terms of injective homomorphisms

A subgroup of a group can also be defined as another abstract group along with an injective homomorphism (or embedding) from that abstract group to the given group. Here, the other abstract group can be naturally identified via its image under the homomorphism, which is the subgroup in a more literal sense.

Often, when we want to emphasize the subgroup not just as an abstract group but in its role as a subgroup, we use the term embedding and think of it as an injective homomorphism.

Equivalence of subgroups

Given a subgroup $H_{1}\leq G_{1}$ and a subgroup $H_{2}\leq G_{2}$, we say that these two subgroups are equivalent if there is an isomorphism $\sigma$ from $G_{1}$ to $G_{2}$ such that $H_{1}$ maps to $H_{2}$ under that isomorphism.

In particular, if $G_{1}=G_{2}=G$, then $H_{1}$ and $H_{2}$ are equivalent as subgroups if there is an automorphism of $G$ under which $H_{1}$ maps to $H_{2}$.

This notion of equivalence of subgroups is important when dealing with and defining the notion of subgroup property.