Subgroup

Definition

A subgroup of a group $G$ is a subset $H$ of $G$ such that $H$ itself is a group under the same binary operation defined on $G$.

If $G$ is a group with operation $*$, then a nonempty subset $H \subseteq G$ is a subgroup if:

1. For all $a, b \in H$, the element $a * b \in H$
2. For every $a \in H$, the inverse $a^{-1} \in H$

Equivalently, $H$ is a subgroup if it is nonempty and closed under the operation and inverses.
A commonly used compact test is:

• $H \neq \varnothing$
• For all $a, b \in H$, $a * b^{-1} \in H$

This is called the subgroup test.

Example:

• Let $G = (\mathbb{Z}, +)$, the group of integers under addition.
• Let $H = 2\mathbb{Z} = \{\dots, -4, -2, 0, 2, 4, \dots\}$.
• Then $H$ is a subgroup of $\mathbb{Z}$ because the sum of even integers is even and the additive inverse of an even integer is also even.

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Main Content

1. First Concept: Basic Properties of Subgroups

• A subgroup must contain the identity element of the parent group. This is automatic once the set is nonempty and closed under the subgroup condition.
• Every element in a subgroup must have its inverse inside the subgroup, so the subgroup is closed under “undoing” the group operation.

A few important consequences follow from the definition:

• If $H$ is a subgroup of $G$, then the operation on $H$ is simply the same operation as in $G$, restricted to elements of $H$.
• A subgroup is itself a group, so it satisfies associativity, identity, and inverse properties.
• The smallest subgroup of any group $G$ is the trivial subgroup $\{e\}$, where $e$ is the identity element.

Example:

• In $(\mathbb{Z}, +)$, the subset $\{0\}$ is a subgroup because $0+0=0$, and the inverse of $0$ is $0$.

Another example:

• In the multiplicative group of nonzero real numbers $(\mathbb{R}^\times, \cdot)$, the set of positive real numbers $\mathbb{R}_{>0}$ is a subgroup because:
• the product of positive numbers is positive,
• the inverse of a positive number is positive,
• $1$ is included.

2. Second Concept: Subgroup Test

• To check whether a subset is a subgroup, it is usually enough to verify the subgroup test rather than checking all group axioms separately.
• The most efficient form is: a nonempty subset $H$ of $G$ is a subgroup if for all $a, b \in H$, the element $ab^{-1}$ also lies in $H$.

Why this works:

• Closure under $ab^{-1}$ implies closure under inverses and the operation itself.
• If $a, b \in H$, then by taking $b = e$ or using the subgroup structure carefully, one can derive the identity and inverses.

This test is especially useful when working with concrete sets.

Example:

• Let $G = (\mathbb{R}, +)$ and $H = \{x \in \mathbb{R} : x \text{ is an integer multiple of } 3\}$.
• If $a, b \in H$, then $a-b \in H$.
• Since the operation is addition, the subgroup test becomes: if $a, b \in H$, then $a + (-b) \in H$, which is true.
• Hence $H$ is a subgroup of $\mathbb{R}$ under addition.

Non-example:

• The set of positive integers under addition is not a subgroup of $(\mathbb{Z}, +)$ because it does not contain inverses. For example, $3$ is in the set, but $-3$ is not.

3. Third Concept: Types and Examples of Subgroups

• Subgroups can be classified in several useful ways, such as trivial subgroup, proper subgroup, cyclic subgroup, and normal subgroup.
• Different examples help illustrate how subgroup ideas appear in arithmetic, geometry, and symmetry.

Important types:

Trivial subgroup

• The subgroup containing only the identity element.
• Example: $\{0\}$ in $(\mathbb{Z}, +)$

Improper subgroup

• The entire group $G$ is also always a subgroup of itself.
• Example: $\mathbb{Z}$ is a subgroup of $\mathbb{Z}$

Proper subgroup

• Any subgroup $H$ such that $H \neq G$.
• Example: $2\mathbb{Z}$ is a proper subgroup of $\mathbb{Z}$

Cyclic subgroup

• Generated by a single element $g$, written $\langle g \rangle$
• It consists of all powers of $g$ in multiplicative groups or all integer multiples of $g$ in additive groups.
• Example: In $(\mathbb{Z}, +)$, the subgroup generated by $3$ is $\langle 3 \rangle = \{3k : k \in \mathbb{Z}\}$

Normal subgroup

• A subgroup with an additional symmetry condition important in quotient groups.
• Although every normal subgroup is a subgroup, not every subgroup is normal.

A simple relationship diagram:

Group G
├── Trivial subgroup {e}
├── Proper subgroups
│   ├── Cyclic subgroups
│   └── Other subgroups
└── The whole group G

This shows that a group contains many possible subgroups, each revealing structure inside the larger system.

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Working / Process

1. Identify the parent group

• First determine the group $G$ you are working within, including its operation.
• Example: $(\mathbb{Z}, +)$, $(\mathbb{R}^\times, \cdot)$, or a symmetry group.

2. Select the subset

• Choose the subset $H$ you want to test.
• Make sure you understand exactly which elements are included.

3. Apply the subgroup test

• Check that the subset is nonempty.
• Verify closure under the operation and inverses, or use the compact test $ab^{-1} \in H$.
• If the test fails, the subset is not a subgroup; if it passes, it is a subgroup.

Example process:

• Let $H = \{x \in \mathbb{R} : x > 0\}$ inside $(\mathbb{R}, +)$
• This is not a subgroup because:
• it is not closed under addition inverses
• if $x > 0$, then $-x < 0$, so the inverse is not in $H$

Another example:

• Let $H = \{x \in \mathbb{R} : x = 5n, n \in \mathbb{Z}\}$ inside $(\mathbb{R}, +)$
• $H$ is nonempty because $0 \in H$
• If $a=5m$ and $b=5n$, then $a-b=5(m-n)\in H$
• Therefore $H$ is a subgroup

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Advantages / Applications

• Subgroups simplify the study of large groups by isolating smaller structures that are easier to analyze.
• They are essential in understanding symmetry, including geometric symmetries, permutations, and transformation groups.
• Subgroups are the foundation for advanced topics such as cosets, quotient groups, group homomorphisms, and the classification of algebraic structures.

Additional applications include:

Number theory

• Subgroups help describe divisibility patterns and modular arithmetic.

Physics and chemistry

• Symmetry subgroups are used to study molecular symmetry and conservation laws.

Computer science

• Group-based methods appear in cryptography, coding theory, and algorithmic symmetry detection.

Geometry

• Rotation and reflection subgroups explain shapes and motions.

Examples in application:

• The rotational symmetries of a square form a subgroup of its full symmetry group.
• The set of even integers forms a subgroup that is useful in modular arithmetic and parity arguments.
• Subgroups of permutation groups help model rearrangements and combinatorial structures.

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Summary

• A subgroup is a subset of a group that is itself a group under the same operation.
• The subgroup test provides a fast way to verify whether a subset is a subgroup.
• Common examples include trivial, proper, cyclic, and normal subgroups.
• Important terms to remember: group, subgroup, identity element, inverse, closure, subgroup test, cyclic subgroup, normal subgroup