The Central Idea
Suppose
\[ (G,\star) \]is a group.
A subset \(H\subseteq G\) may itself form a group under the same operation. If it does, then \(H\) is called a subgroup of \(G\).
At the same time, many groups arise naturally from collections of transformations that preserve some object. These are symmetry groups.
The two ideas fit together because a large symmetry group often contains smaller symmetry subgroups.
What Is a Subgroup?
Let \((G,\star)\) be a group.
A subset
\[ H\subseteq G \]is a subgroup if
\[ (H,\star) \]is itself a group using the same operation.
We write
\[ \boxed{H\le G} \]to mean that \(H\) is a subgroup of \(G\).
The Operation Does Not Change
A subgroup uses the same operation as the larger group.
If
\[ (G,\star) \]is the original group, then a subgroup has the form
\[ (H,\star), \]not
\[ (H,\circ) \]with some unrelated operation.
The subgroup inherits its algebraic behavior from the larger structure.
What Must a Subgroup Satisfy?
Since \(H\) must be a group, it must satisfy:
- closure under the group operation;
- associativity;
- containment of the identity;
- containment of inverses.
However, associativity usually does not need to be re-proved because it is inherited from \(G\).
Why Associativity Is Inherited
Suppose \(a,b,c\in H\).
Since
\[ H\subseteq G, \]we also have
\[ a,b,c\in G. \]Because \(G\) is a group,
\[ (a\star b)\star c = a\star(b\star c). \]Therefore associativity automatically holds for elements of \(H\).
A Practical Subgroup Checklist
To show that \(H\subseteq G\) is a subgroup, it is enough to verify:
- \(H\) is nonempty;
- if \(a,b\in H\), then \(a\star b\in H\);
- if \(a\in H\), then \(a^{-1}\in H\).
Associativity is inherited from \(G\), and the identity will follow from the subgroup conditions.
The One-Step Subgroup Test
In a group \((G,\star)\), a nonempty subset \(H\subseteq G\) is a subgroup if
\[ \boxed{ a,b\in H \Longrightarrow a\star b^{-1}\in H }. \]This is often called the subgroup test.
It combines closure and inverse behavior into one condition.
Why the Subgroup Test Works
Suppose \(H\) is nonempty and
\[ a,b\in H \Longrightarrow a\star b^{-1}\in H. \]Choose some \(a\in H\). Then
\[ a\star a^{-1}=e\in H. \]So the identity belongs to \(H\).
Now if \(a\in H\), then using \(e,a\in H\),
\[ e\star a^{-1}=a^{-1}\in H. \]Thus inverses belong to \(H\).
Finally, if \(a,b\in H\), then \(b^{-1}\in H\), so
\[ a\star(b^{-1})^{-1}=a\star b\in H. \]Hence closure also holds.
Example: Even Integers Under Addition
Consider the additive group
\[ (\mathbb{Z},+). \]Let
\[ 2\mathbb{Z} = \{\ldots,-4,-2,0,2,4,\ldots\}. \]This is the set of even integers.
We will show that
\[ \boxed{2\mathbb{Z}\le\mathbb{Z}} \]under addition.
Checking the Even-Integer Subgroup
If
\[ a=2m \]and
\[ b=2n, \]then
\[ a+b=2m+2n=2(m+n), \]which is even.
Also,
\[ -a=-2m=2(-m), \]which is even.
The identity \(0\) is even.
Therefore the even integers form a subgroup of the integers under addition.
Multiples of \(n\) Form Additive Subgroups
More generally, for any positive integer \(n\),
\[ n\mathbb{Z} = \{nk:k\in\mathbb{Z}\} \]is a subgroup of
\[ (\mathbb{Z},+). \]If
\[ a=nm,\qquad b=nk, \]then
\[ a-b=n(m-k), \]which is again a multiple of \(n\).
A Subset Need Not Be a Subgroup
Consider the positive integers
\[ \mathbb{Z}_{>0} \]as a subset of \((\mathbb{Z},+)\).
The additive identity
\[ 0 \]is not in \(\mathbb{Z}_{>0}\).
Also, if \(5\in\mathbb{Z}_{>0}\), its additive inverse
\[ -5 \]is not in the set.
Therefore the positive integers are not a subgroup of the integers under addition.
The Trivial Subgroup
Every group \(G\) has the subgroup containing only the identity:
\[ \boxed{\{e\}\le G}. \]This is called the trivial subgroup.
It is a group because
\[ e\star e=e, \]and \(e\) is its own inverse.
The Whole Group Is a Subgroup
Every group is also a subgroup of itself:
\[ \boxed{G\le G}. \]Therefore every group has at least two obvious subgroup candidates:
\[ \{e\} \]and
\[ G. \]When these are distinct, they are called the trivial and whole-group subgroups.
Proper Subgroups
A subgroup \(H\) is a proper subgroup of \(G\) if
\[ H\le G \]and
\[ H\ne G. \]One common notation is
\[ HSubgroups Share the Same Identity
If \(H\le G\), then the identity element of \(H\) is the same identity element used by \(G\).
A subgroup cannot introduce a different identity for the same operation.
This follows from the uniqueness of identity elements.
Subgroup Inverses Are the Same Inverses
If \(a\in H\le G\), then the inverse of \(a\) in \(H\) is the same element as the inverse of \(a\) in \(G\).
This follows from uniqueness of inverses.
The subgroup does not redefine the inverse operation; it inherits it.
Intersections of Subgroups
If \(H\) and \(K\) are subgroups of \(G\), then
\[ H\cap K \]is also a subgroup of \(G\).
The identity lies in both \(H\) and \(K\), so the intersection is nonempty.
If \(a,b\in H\cap K\), then both \(a\) and \(b\) belong to each subgroup, so
\[ a\star b^{-1} \]belongs to both.
Therefore,
\[ \boxed{H\cap K\le G}. \]Unions of Subgroups Need Not Be Subgroups
By contrast,
\[ H\cup K \]need not be a subgroup.
For example, in \((\mathbb{Z},+)\), consider
\[ 2\mathbb{Z} \]and
\[ 3\mathbb{Z}. \]We have
\[ 2\in2\mathbb{Z} \]and
\[ 3\in3\mathbb{Z}. \]But
\[ 2+3=5, \]and \(5\) belongs to neither \(2\mathbb{Z}\) nor \(3\mathbb{Z}\).
So the union is not closed under addition.
Generated Subgroups
Given an element \(a\in G\), we can consider all elements produced by repeatedly applying the group operation and inverses to \(a\).
The resulting subgroup is written
\[ \boxed{\langle a\rangle}. \]It is called the subgroup generated by \(a\).
Cyclic Groups
A group \(G\) is cyclic if some element \(a\in G\) generates the entire group:
\[ \boxed{G=\langle a\rangle}. \]The element \(a\) is called a generator.
Cyclic groups provide some of the simplest and most important examples of group structure.
Example: The Integers Are Cyclic
Under addition,
\[ (\mathbb{Z},+) \]is generated by \(1\):
\[ \langle1\rangle=\mathbb{Z}. \]Repeated addition of \(1\) produces all positive integers, the identity gives \(0\), and additive inverses produce all negative integers.
Thus
\[ \boxed{\mathbb{Z}=\langle1\rangle}. \]The Integers Also Have Generator \(-1\)
Under addition,
\[ \langle-1\rangle=\mathbb{Z} \]as well.
Therefore a cyclic group may have more than one generator.
What Is a Symmetry?
A symmetry of an object is a transformation that leaves the object indistinguishable from its original configuration.
Depending on the object, symmetries may include:
- rotations;
- reflections;
- translations;
- permutations of equivalent components;
- other structure-preserving transformations.
The symmetries of an object naturally form a group under composition.
Why Symmetries Form a Group
Let \(G\) be the set of symmetries of some object.
Under composition:
- composing two symmetries gives another symmetry;
- composition is associative;
- doing nothing is an identity symmetry;
- every symmetry can be undone by an inverse symmetry.
Therefore the symmetries form a group.
The Identity Symmetry
Every symmetry group contains an identity transformation, often denoted
\[ e \]or
\[ I. \]It leaves every point unchanged.
In geometric language, it can be thought of as a rotation by
\[ 0^\circ. \]Inverse Symmetries
Every symmetry has an inverse transformation that undoes it.
For example, a rotation by
\[ 90^\circ \]is undone by a rotation by
\[ -90^\circ \]or equivalently
\[ 270^\circ. \]A reflection is often its own inverse because applying the same reflection twice returns the object to its original position.
Rotational Symmetry of a Square
A square can be rotated about its center by
\[ 0^\circ,\qquad90^\circ,\qquad180^\circ,\qquad270^\circ \]and remain unchanged as a square.
Let these rotations be
\[ e,\qquad r,\qquad r^2,\qquad r^3. \]Here \(r\) represents a \(90^\circ\) rotation.
The Rotation Subgroup of the Square
The four rotations form a group under composition:
\[ \boxed{ \{e,r,r^2,r^3\} }. \]Since
\[ r^4=e, \]repeated composition cycles back to the identity.
This is a cyclic group generated by \(r\):
\[ \boxed{\langle r\rangle=\{e,r,r^2,r^3\}}. \]Rotation Composition
Composing rotations adds their angles modulo
\[ 360^\circ. \]For example,
\[ r\circ r=r^2, \]corresponding to
\[ 90^\circ+90^\circ=180^\circ. \]Also,
\[ r^3\circ r=r^4=e. \]Full Symmetry Group of a Square
A square has more symmetries than rotations alone.
It also has four reflections.
Altogether, the square has
\[ \boxed{8} \]symmetries.
These form the dihedral group
\[ \boxed{D_4}. \]The Eight Symmetries of a Square
One common description of \(D_4\) is
\[ D_4 = \{e,r,r^2,r^3,s,sr,sr^2,sr^3\}, \]where
\[ r \]is a \(90^\circ\) rotation and
\[ s \]is one chosen reflection.
The other reflections arise by combining \(s\) with rotations.
A Key Dihedral Relation
The square's symmetry group satisfies
\[ r^4=e, \] \[ s^2=e, \]and
\[ \boxed{srs=r^{-1}}. \]This relation captures the fact that reflecting reverses the direction of rotation.
Symmetry Groups Need Not Be Abelian
In the full symmetry group of a square,
\[ rs \]and
\[ sr \]are generally different transformations.
Therefore
\[ \boxed{D_4\text{ is nonabelian}}. \]The order in which transformations are performed matters.
Rotations Alone Form a Subgroup
The rotation symmetries
\[ R=\{e,r,r^2,r^3\} \]are contained in
\[ D_4. \]They are closed under composition, contain the identity, and contain inverses.
Therefore,
\[ \boxed{R\le D_4}. \]A Reflection Can Generate a Two-Element Subgroup
If \(s\) is a reflection, then
\[ s^2=e. \]Therefore
\[ \{e,s\} \]is closed under composition.
Thus
\[ \boxed{\{e,s\}\le D_4}. \]Symmetry Groups of Regular Polygons
A regular \(n\)-gon has
\[ n \]rotational symmetries and
\[ n \]reflection symmetries.
Therefore its full symmetry group has
\[ \boxed{2n} \]elements.
This group is commonly denoted
\[ D_n. \]Rotational Subgroup of a Regular \(n\)-gon
The rotations alone form a cyclic subgroup of order \(n\).
If \(r\) is rotation by
\[ \frac{360^\circ}{n}, \]then
\[ \langle r\rangle = \{e,r,r^2,\ldots,r^{n-1}\}. \]Since
\[ r^n=e, \]the sequence returns to the identity after \(n\) rotations.
Permutations as Symmetries
A permutation is a rearrangement of a set of objects.
If an object has labeled vertices, edges, positions, or components, each symmetry can be described by how it permutes those labels.
This creates a direct connection between geometric symmetry groups and permutation groups.
The Symmetric Group
The set of all permutations of \(n\) objects forms a group under composition.
This group is called the symmetric group and is written
\[ \boxed{S_n}. \]Since there are
\[ n! \]permutations of \(n\) objects,
\[ |S_n|=n!. \]Example: \(S_3\)
The group
\[ S_3 \]contains all permutations of three objects.
Therefore,
\[ |S_3|=3!=6. \]This group is nonabelian.
It is also structurally equivalent to the full symmetry group of an equilateral triangle.
Symmetry of an Equilateral Triangle
An equilateral triangle has:
- three rotations;
- three reflections.
Therefore its full symmetry group has
\[ 6 \]elements.
This group is usually denoted
\[ D_3, \]and it is isomorphic to
\[ S_3. \]Different Objects Can Have the Same Group Structure
Two groups can have different kinds of elements yet share the same algebraic structure.
For example:
\[ D_3 \]can be viewed as geometric symmetries of a triangle, while
\[ S_3 \]can be viewed as permutations of three labels.
Their elements look different, but the pattern of composition is the same.
This idea will later be formalized through isomorphisms.
Symmetry Turns Geometry into Algebra
Geometry asks which transformations preserve a figure.
Algebra asks how those transformations combine.
Group theory links the two questions by treating each symmetry as an algebraic element and composition as the group operation.
In this way,
\[ \boxed{\text{geometric motion becomes algebraic structure}}. \]Symmetry Groups Encode Constraints
A symmetry group tells us which transformations preserve the essential structure of an object.
This makes symmetry groups useful far beyond elementary geometry.
They appear in:
- crystallography;
- chemistry;
- physics;
- number theory;
- geometry;
- combinatorics;
- coding and information theory.
Finite Subgroups and Operation Tables
A finite subgroup can be represented with an operation table just like any finite group.
The table will:
- contain only subgroup elements;
- contain the shared identity;
- contain every subgroup element exactly once in each row and column;
- remain closed within the subgroup.
This gives a practical way to recognize small subgroup structures.
Example: A Two-Element Group
Let
\[ H=\{e,s\} \]where
\[ s^2=e. \]Its operation table is
| \(\star\) | \(e\) | \(s\) |
|---|---|---|
| \(e\) | \(e\) | \(s\) |
| \(s\) | \(s\) | \(e\) |
The element \(s\) is self-inverse.
This is the structure of any group with exactly two elements.
Order of a Finite Group
The order of a finite group \(G\) is the number of elements it contains.
It is written
\[ \boxed{|G|}. \]For example,
\[ |D_4|=8 \]and
\[ |S_3|=6. \]Order of an Element
The order of an element \(a\) in a finite group is the smallest positive integer \(n\) such that
\[ \boxed{a^n=e}. \]For additive groups, the analogous notation is
\[ na=0. \]The identity itself has order \(1\).
Example: Orders of Square Rotations
In the rotational symmetry group of a square:
- \(e\) has order \(1\);
- \(r\) has order \(4\);
- \(r^2\) has order \(2\);
- \(r^3\) has order \(4\).
For example,
\[ (r^2)^2=r^4=e, \]so \(r^2\) has order \(2\).
Generated Subgroups and Element Order
If an element \(a\) has finite order \(n\), then
\[ \langle a\rangle = \{e,a,a^2,\ldots,a^{n-1}\}. \]Therefore,
\[ \boxed{|\langle a\rangle|=n}. \]The order of the element equals the size of the cyclic subgroup it generates.
Subgroups Reveal Internal Structure
A group need not be understood all at once.
Its subgroups can reveal:
- repeating cycles;
- restricted symmetry families;
- invariant transformations;
- smaller algebraic systems inside larger ones;
- relationships among generators and operations.
Subgroups are therefore one of the basic tools for decomposing and analyzing group structure.
Common Error: Checking Only That \(H\subseteq G\)
Being a subset is not enough.
For example,
\[ \mathbb{Z}_{>0}\subset\mathbb{Z}, \]but the positive integers are not a subgroup of \((\mathbb{Z},+)\).
The subgroup axioms must still hold within the subset.
Common Error: Using a Different Operation
A subgroup must use the same operation as the parent group.
If \(G\) is studied under addition, then a subgroup must also be studied under addition.
A subset that forms a group under some different operation is not thereby a subgroup of the original group.
Common Error: Forgetting the Identity
Every subgroup must contain the identity of the larger group.
Therefore any subset that omits the identity can immediately be rejected as a subgroup.
This is often the fastest first check.
Common Error: Forgetting Inverses
A subset may be closed under the operation yet fail to contain inverses.
For example,
\[ \mathbb{Z}_{\ge0} \]is closed under addition, but the inverse of \(5\) is
\[ -5, \]which is not in the set.
Therefore closure alone does not establish subgroup structure.
Common Error: Assuming All Symmetry Groups Are Abelian
Rotations of a regular polygon form an abelian subgroup, but the full symmetry group may not be abelian.
In a square,
\[ rs\ne sr \]in general.
Therefore the full dihedral group is noncommutative.
Common Error: Confusing Group Order with Element Order
The order of a group is
\[ |G|, \]the number of elements in the group.
The order of an element \(a\) is the smallest positive integer \(n\) such that
\[ a^n=e. \]These are related concepts but not the same quantity.
A Reliable Subgroup Procedure
- Confirm that \(H\subseteq G\).
- Confirm that \(H\) is nonempty.
- Check that the identity of \(G\) belongs to \(H\).
- Check closure under the group operation.
- Check that inverses remain in \(H\).
- Use inherited associativity rather than re-proving it.
- When convenient, replace separate closure and inverse checks with the subgroup test \(a\star b^{-1}\in H\).
A Reliable Symmetry-Group Procedure
- Identify the object whose symmetries are being studied.
- List transformations that preserve the object.
- Use composition as the operation.
- Identify the identity transformation.
- Determine the inverse of each symmetry.
- Check whether order matters when composing transformations.
- Look for rotation-only or reflection-generated subgroups.
- Represent the transformations as permutations when useful.
Example: Full Subgroup Check
Let
\[ G=(\mathbb{Z},+) \]and
\[ H=5\mathbb{Z}. \]We use the additive subgroup test.
Let
\[ a=5m,\qquad b=5n. \]Since the additive inverse of \(b\) is \(-b\),
\[ a+(-b) = 5m-5n = 5(m-n). \]Because
\[ m-n\in\mathbb{Z}, \]we have
\[ a-b\in5\mathbb{Z}. \]Therefore,
\[ \boxed{5\mathbb{Z}\le\mathbb{Z}}. \]Practice 1: Even Integers
Explain why
\[ 2\mathbb{Z} \]is a subgroup of
\[ (\mathbb{Z},+). \]Practice 2: Positive Integers
Are the positive integers a subgroup of
\[ (\mathbb{Z},+)? \]Explain.
Practice 3: Multiples of 4
Determine whether
\[ 4\mathbb{Z} \]is a subgroup of
\[ (\mathbb{Z},+). \]Practice 4: Trivial Subgroup
State the trivial subgroup of
\[ (\mathbb{Z},+). \]Practice 5: Square Rotations
How many rotational symmetries does a square have?
Practice 6: Full Square Symmetry Group
How many elements are in
\[ D_4? \]Practice 7: Reflection Inverse
If \(s\) is a reflection satisfying
\[ s^2=e, \]what is \(s^{-1}\)?
Practice 8: Order of a Rotation
In the symmetry group of a square, \(r\) is rotation by
\[ 90^\circ. \]What is the order of \(r\)?
Practice 9: Symmetric Group
How many elements are in
\[ S_4? \]Practice 10: Identify the Error
A student says:
“The rotations of a square are not a subgroup of \(D_4\) because they do not include the reflections.”
Explain the error.
Practice Answers
Practice 1
If
\[ a=2m,\qquad b=2n, \]then
\[ a-b=2(m-n), \]which is even.
Therefore,
\[ \boxed{2\mathbb{Z}\le\mathbb{Z}}. \]Practice 2
No. The identity \(0\) is not positive, and the additive inverse of a positive integer is negative.
Therefore the positive integers are not a subgroup of \((\mathbb{Z},+)\).
Practice 3
Yes. If
\[ a=4m,\qquad b=4n, \]then
\[ a-b=4(m-n)\in4\mathbb{Z}. \]Therefore,
\[ \boxed{4\mathbb{Z}\le\mathbb{Z}}. \]Practice 4
The identity of \((\mathbb{Z},+)\) is \(0\), so the trivial subgroup is
\[ \boxed{\{0\}}. \]Practice 5
A square has
\[ \boxed{4} \]rotational symmetries:
\[ 0^\circ,\quad90^\circ,\quad180^\circ,\quad270^\circ. \]Practice 6
The full symmetry group of a square contains four rotations and four reflections.
Therefore,
\[ \boxed{|D_4|=8}. \]Practice 7
Since
\[ s^2=e, \]the reflection undoes itself.
Therefore,
\[ \boxed{s^{-1}=s}. \]Practice 8
Four \(90^\circ\) rotations return to the identity:
\[ r^4=e. \]Therefore the order of \(r\) is
\[ \boxed{4}. \]Practice 9
The symmetric group \(S_4\) contains
\[ 4!=24 \]permutations.
Therefore,
\[ \boxed{|S_4|=24}. \]Practice 10
A subgroup does not need to contain every element of the larger group.
The rotations
\[ \{e,r,r^2,r^3\} \]are closed under composition, contain the identity, and contain inverses.
Therefore they form a subgroup of \(D_4\) even though they omit the reflections.
What to Remember
A subgroup is a subset that remains a group under the same operation:
\[ \boxed{H\le G}. \]A practical subgroup test is
\[ \boxed{ a,b\in H \Longrightarrow a\star b^{-1}\in H }. \]Symmetry groups arise when transformations preserving an object are composed.
- rotations can form cyclic subgroups;
- reflections are often self-inverse;
- regular \(n\)-gons have dihedral symmetry groups of order \(2n\);
- permutations form symmetric groups \(S_n\);
- subgroups expose smaller repeated structures inside larger groups.
Symmetry groups demonstrate one of the central themes of abstract algebra: very different objects can share the same underlying structural pattern.
Where We Go Next
Groups organize one operation together with identity and inverse structure. Many familiar number systems carry two interacting operations, usually addition and multiplication.
The next lesson, Rings, Integral Domains, and Fields, develops algebraic systems built from two operations and explains how increasingly strong properties distinguish rings, integral domains, and fields.