Subgroup

A subgroup is a subset of a group $𝐻 \subseteq 𝐺$ such that $𝐻$ is a group under the same operations #m/def/group , i.e. $𝐻$ is

closed under the group operation
contains the inverse of every element

Tests for subgroups

Let $𝐺$ be a group and $𝐻 \subseteq 𝐺$ be a inhabited subset. Additionally we define predicate $𝑝$ so that $𝐻 = {𝑥 \in 𝐺 ∣ 𝑝 (𝑥)}$ . The following hold:¹

One step subgroup test

Theorem. Iff $𝑎 𝑏 - 1 \in 𝐻$ whenever $𝑎, 𝑏 \in 𝐻$ , then $𝐻$ is a subgroup of $𝐺$ . #m/thm/group

Proof

Since $𝐻$ is inhabited there exists $𝑥 \in 𝐻$ , then with $𝑎 = 𝑏 = 𝑥$ clearly $𝑒 \in 𝐻$ , $𝐻$ must be closed under inversion, since letting $𝑎 = 𝑒$ for any $𝑏 \in 𝐻$ we have $𝑒 𝑏 - 1 = 𝑏 - 1 \in 𝐻$ . Now we can show that $𝐻$ is closed in general: For any $𝑎, 𝑏 \in 𝐻$ we have $𝑏 - 1 \in 𝐻$ and therefore $𝑎 𝑏 = 𝑎 (𝑏 - 1) - 1 \in 𝐻$ .

Application

Show $𝑝 (𝑒)$
Assume $𝑝 (𝑎)$ and $𝑝 (𝑏)$
Prove $𝑝 (𝑎 𝑏 - 1)$

Two step subgroup test

Theorem. Iff $𝐻$ is closed under the binary operation and under the inverse, then $𝐻$ is a subgroup of $𝐺$ . #m/thm/group

Proof

Since $𝐻$ is inhabited there exists $𝑥$ in $𝐻$ , thus $𝑥 - 1 \in 𝐻$ and thus $𝑒 = 𝑥 𝑥 - 1 \in 𝐻$ . Thence $𝐻$ is a subgroup of $𝐺$ .

Application

Much the same as above, but with

Prove $𝑝 (𝑎 - 1)$ and $𝑝 (𝑎 𝑏)$ .

Finite subgroup test

Theorem. $𝐻$ is finite and closed under the binary operation, then it is a subgroup of $𝐺$ . #m/thm/group

Proof

Take any $𝑥 \in 𝐻$ . Since $𝐻$ is closed we may construct a sequence $(𝑥 𝑛) \infty 𝑛 = 1 \in 𝐻$ . Since $𝐻$ is finite, by the Pigeonhole principle the sequence must have repeated elements, so that for some $1 < 𝑖 < 𝑗$ we have $𝑥 𝑖 = 𝑥 𝑗$ . Then $𝑎 𝑖 - 𝑗 = 𝑒$ and hence $𝑎 𝑎 𝑖 - 𝑗 - 1 = 𝑒$ , so $𝑎 - 1 = 𝑎 𝑖 - 𝑗 - 1 \in 𝐻$ . Therefore $𝐻$ is closed under the inverse and the binary operation, and is thus a subgroup of $𝐺$ by the Two step subgroup test.

Examples of subgroups

For any element $𝑎 \in 𝐺$ we can generate a Cyclic subgroup $⟨ 𝑎 ⟩ = {\dots, 𝑎 - 2, 𝑎 - 1, 𝑒, 𝑎, 𝑎 2, \dots}$ .
The Centre of a group is a subgroup containing elements that commute with all elements.
Similarly the Centralizer in a group is a subgroup containing elements that commute with a given element.
Torsion subgroup of an abelian group
The order of a subgroup divides the order of a group

Properties

The intersection of subgroups is a subgroup
Subgroups may be Conjugate subgroups if they can be derived from each other by conjugation. A subgroup with no other conjugate subgroups is called a Normal subgroup.

#state/tidy | #lang/en | #SemBr

2017, Contemporary Abstract Algebra, pp. 62–64 ↩

Subgroup

Tests for subgroups

One step subgroup test

Two step subgroup test

Finite subgroup test

Examples of subgroups

Properties

Footnotes