Condition for a subset to be a subgroup

Dependencies:

1. Group
2. Subgroup
3. Conditions for a subset to be a subgroup

$H$ is a subgroup of $G$ iff $H \neq \phi$ and $(h_1, h_2 \in H \Rightarrow h_1 h_2^{-1} \in H)$.

Proof

We will prove both sides of the implication.

Part 1

Let $H$ be a group. Then $H \neq \phi$.

$h_1, h_2 \in H \Rightarrow h_1, h_2^{-1} \in H \Rightarrow h_1 h_2^{-1} \in H$.

Therefore, if $H$ is a subgroup of $G$, then $H \neq \phi$ and $(h_1, h_2 \in H \Rightarrow h_1 h_2^{-1} \in H)$.

Part 2

$H \neq \phi \Rightarrow \exists h \in H$.

In $(h_1, h_2 \in H \Rightarrow h_1 h_2^{-1} \in H)$:

• Substitute $(h_1=h, h_2=h)$ to get $\operatorname{id}(G) \in H$.
• Substitute $(h_1=\operatorname{id}(G), h_2=h)$ to get $h^{-1} \in H$.
• $(h_1=h_1, h_2=h_2^{-1})$ to get $h_1 h_2 \in H$.

By the above 3 properties, $H$ is a subgroup of $G$.

Dependency for:

1. Second isomorphism theorem
2. Alternating Group
3. Correspondence theorem
4. Homomorphic mapping of subgroup of domain is subgroup of codomain
5. Conditions for a subset of a ring to be a subring
6. Condition for being a subspace
7. F[x]/p(x): A ring

Info:

• Depth: 3
• Number of transitive dependencies: 5

Transitive dependencies:

1. Group
2. Identity of a group is unique
3. Subgroup
4. Inverse of a group element is unique
5. Conditions for a subset to be a subgroup