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Introduction to Groups in Abstract Algebra

Learn what groups are in abstract algebra, including axioms, notation, examples, subgroups, cyclic groups, Cayley tables, homomorphisms, and key consequences.

A group is a set equipped with an operation that combines any two elements and satisfies four rules. Groups provide a common language for studying arithmetic, symmetries, permutations, matrices, and many other mathematical structures. This subject is part of abstract algebra: instead of focusing only on particular numbers, we study the properties shared by many systems.

What Is a Group?

A binary operation on a set G is a rule that takes two elements of G and produces one element of G. We write a group as (G, *) or (G, ·), where * or · denotes the operation.

The operation does not have to be ordinary addition or multiplication. It could be function composition, permutation composition, rotation of a geometric object, or addition modulo a fixed integer.

The Four Group Axioms

• Closure: For every a, b in G, the result a * b is also in G.
• Associativity: For every a, b, c in G, (a * b) * c = a * (b * c). Parentheses may be moved, but the order of the elements is unchanged.
• Identity: There is an element e in G such that e * a = a * e = a for every a in G.
• Inverses: For every a in G, there is an element a^-1 in G such that a * a^-1 = a^-1 * a = e.

Group axiom checklist

Axiom: Closure | What to check: Combining two members stays in the set | Additive: a + b is in G | Multiplicative: ab is in G | Typical failure: The result leaves the set

Axiom: Associativity | What to check: Parentheses do not affect the result | Additive: (a+b)+c=a+(b+c) | Multiplicative: (ab)c=a(bc) | Typical failure: Subtraction or division

Axiom: Identity | What to check: One element changes nothing | Additive: 0+a=a | Multiplicative: 1a=a | Typical failure: No suitable neutral element

Axiom: Inverses | What to check: Every element can be combined with another to reach the identity | Additive: a+(-a)=0 | Multiplicative: aa^-1=1 | Typical failure: Some element has no inverse in the set

Notation: Additive and Multiplicative Groups

In additive notation, the operation is written as +. The identity is written 0, and the inverse of a is written -a. The repeated operation is expressed as a multiple: na means adding a to itself n times.

In multiplicative notation, the operation is written by juxtaposition, such as ab, or with ·. The identity is commonly 1 or e, and the inverse is a^-1. Repeated multiplication is expressed using powers, such as aⁿ.

These notational choices do not change the underlying ideas. For example, the additive inverse of 5 is -5, while the multiplicative inverse of 5 is 1/5 when that number belongs to the group.

How to Verify the Group Axioms

1. State the set and operation precisely.
2. Check closure with arbitrary elements, not only selected examples.
3. Prove associativity, or explain that it is inherited from a known associative operation.
4. Find the identity and verify it works on both sides.
5. For an arbitrary element, find an inverse and verify that it belongs to the set.

For (Z, +), closure follows because the sum of two integers is an integer. Addition is associative, 0 is the identity, and every integer a has the integer inverse -a. Therefore the integers under addition form a group.

A single failed axiom is enough to prove that a structure is not a group. Closure alone is not enough, and checking a few examples cannot replace a general proof.

Examples and Nonexamples

Set and operation: Z under addition | Group? Yes | Reason: All four axioms hold | Abelian? Yes

Set and operation: N under addition | Group? No | Reason: Positive natural numbers generally lack additive inverses | Abelian? Not applicable

Set and operation: Z under multiplication | Group? No | Reason: Most integers do not have integer multiplicative inverses | Abelian? Not applicable

Set and operation: Nonzero rational, real, or complex numbers under multiplication | Group? Yes | Reason: Every nonzero element has a reciprocal | Abelian? Yes

Set and operation: All real numbers under multiplication | Group? No | Reason: 0 has no multiplicative inverse | Abelian? Not applicable

Set and operation: All square matrices under multiplication | Group? No | Reason: Singular matrices have no inverse | Abelian? Usually no

Set and operation: Permutations under composition | Group? Yes | Reason: Composition, identity permutation, and inverse permutations satisfy the axioms | Abelian? Usually no

Finite, Infinite, Abelian, and Nonabelian Groups

The order of a group, written |G|, is the number of elements in the group. A group is finite if this number is finite and infinite otherwise. For example, Z₆ has order 6, while Z is infinite.

A group is abelian, or commutative, if a * b = b * a for every pair of elements. Integer addition and addition modulo n are abelian. A group that is not abelian is called nonabelian. In a nonabelian group, operation order matters: ab may differ from ba.

Associativity and commutativity are different. Associativity changes parentheses; commutativity changes the order of elements. Groups require associativity, but they do not require commutativity.

Integers Modulo n

The group (Z_n, +) consists of residue classes modulo n, commonly represented by 0, 1, ..., n-1, with addition reduced modulo n. For example, in Z₆, 4 + 5 = 3 because 9 is congruent to 3 modulo 6.

This is a finite cyclic group. The element 1 generates every residue: 0, 1, 2, 3, 4, 5, and then 0 again. The multiplicative group of units modulo 8 is U(8) = {1, 3, 5, 7}. It excludes residues such as 2, 4, and 6 because they have no multiplicative inverse modulo 8.

Subgroups

A subgroup H of a group G, written H ≤ G, is a subset that forms a group using the operation inherited from G. A proper subgroup is a subgroup different from the whole group. The trivial subgroup contains only the identity.

A useful subgroup test says that a nonempty subset H of G is a subgroup if, for every a, b in H, the element ab^-1 is also in H. This single condition guarantees closure under products and inverses. In additive notation, the condition becomes a - b in H.

The even integers form a subgroup: 2Z ≤ Z under addition. If a and b are even, then a-b is even. More generally, the multiples of any fixed integer m form the subgroup mZ.

Cyclic Groups and Generators

A group is cyclic if one element generates the entire group. Such an element is called a generator. The subgroup generated by g is written <g>.

In additive notation, <g> = {ng : n in Z}, meaning all integer multiples of g. In multiplicative notation, <g> = {gⁿ : n in Z}, meaning all integer powers of g. Thus repeated addition and repeated multiplication are two versions of the same pattern.

The group Z is cyclic, generated by 1 and also by -1. The group Z_n is cyclic, generated by 1. Not every element generates a cyclic group: in Z₆, 2 generates only {0, 2, 4}, not all six elements.

Order of an Element

The order of an element g is the least positive integer k for which g^k = e, if such an integer exists. In additive notation, it is the least positive k for which kg = 0.

For example, in Z₆, the order of 2 is 3 because 2 + 2 + 2 = 0 modulo 6. The subgroup generated by an element contains exactly the repeated powers or multiples of that element. If a finite group has order n, Lagrange's theorem states that the order of every element divides n.

Concept: Order of a group | Meaning: Number of elements in the group | Notation: |G| | Example: |Z₆|=6

Concept: Order of an element | Meaning: Least positive repetition returning to the identity | Notation: ord(g) | Example: ord(2)=3 in Z₆

Concept: Order of a cyclic subgroup | Meaning: Number of elements generated by one element | Notation: |<g>| | Example: |<2>|=3 in Z₆

Cayley Tables

A Cayley table records the result of every pairwise operation in a finite group. For Z₄ under addition modulo 4:

+ mod 4:   0 | 1 | 2 | 3

0: 0 | 1 | 2 | 3

1: 1 | 2 | 3 | 0

2: 2 | 3 | 0 | 1

3: 3 | 0 | 1 | 2

The identity 0 appears unchanged across its row and column. The inverse of an element is found where its row or column produces 0: 1 and 3 are inverses, while 0 and 2 are their own inverses. Every row and column contains every group element exactly once. This is the Latin-square property, which follows from cancellation.

Permutations and Geometric Symmetries

A permutation is a rearrangement of a finite set. All permutations of three objects form the symmetric group S₃ under composition. Composition is associative because it is composition of functions, the identity permutation does nothing, and every permutation has a reverse permutation.

Use the common convention that the rightmost permutation acts first. For instance, applying a swap of positions 1 and 2 followed by a swap of positions 2 and 3 need not give the same result as reversing those two swaps. Therefore S₃ is nonabelian.

The symmetries of an equilateral triangle form the dihedral group D₃. It has three rotations and three reflections, for six elements total. A rotation followed by a reflection can differ from the same reflection followed by the rotation, so the group is nonabelian.

Triangle symmetry composition: Let r be a 120-degree rotation and s a reflection. The relations are r³=e, s²=e, and sr=r^-1s. Consequently, generally sr ≠ rs.

Elements: e, r, r², s, sr, sr²

Key lesson: The order of transformations matters, making D₃ a concrete nonabelian group.

Homomorphisms and Isomorphisms

A homomorphism is a function between groups that preserves their operations. If f: G → H is a homomorphism, then f(ab)=f(a)f(b) in multiplicative notation, or f(a+b)=f(a)+f(b) in additive notation.

The kernel is the set of elements mapped to the identity: ker(f)={g in G : f(g)=e}. The image is the set of values actually produced: im(f)={f(g):g in G}. For reduction modulo n, f: Z → Z_n given by f(k)=k mod n is a homomorphism. Its kernel is nZ, and its image is all of Z_n.

An isomorphism is a bijective homomorphism. Isomorphic groups may contain different kinds of objects or use different notation, but they have exactly the same group structure. An isomorphism preserves identities, inverses, products, element orders, and subgroup relationships.

Consequences of the Group Axioms

• Identity is unique: A group cannot have two different identity elements. If both e and f act as identities, then e=e*f=f.
• Inverses are unique: Each element has only one inverse. If both b and c invert a, then b=b(ac)=(ba)c=c.
• Cancellation holds: If ab=ac, then b=c. Similarly, ba=ca implies b=c. Multiply by the appropriate inverse to cancel.
• Equations can be solved: From ax=b, multiply on the left by a^-1 to obtain x=a^-1b. In a nonabelian group, preserve the side on which multiplication occurs.
• Finite-order relationship: For a finite group, the order of an element divides the order of the group. This is a central consequence known as part of Lagrange's theorem.

Common Mistakes and Troubleshooting

• Do not assume closure alone creates a group; check all four axioms.
• Do not include zero in the multiplicative group of real numbers. Use the nonzero real numbers.
• Do not confuse associativity with commutativity.
• Do not automatically use subtraction or division as group operations; they generally fail associativity.
• A subset containing the identity is not automatically a subgroup. It must also satisfy the subgroup test.
• In a cyclic group, not every element is a generator. Its order must equal the order of the whole group.
• State a convention when composing permutations. With the usual rightmost-first convention, changing the order of composition can change the result.
• Distinguish the order of a group from the order of an element.

Summary

A group is a set with a binary operation satisfying closure, associativity, identity, and inverses. Groups may be additive or multiplicative, finite or infinite, abelian or nonabelian. Important examples include integers under addition, modular addition groups, nonzero numbers under multiplication, permutation groups, and geometric symmetry groups. Subgroups, cyclic groups, generators, element orders, Cayley tables, homomorphisms, kernels, images, and isomorphisms provide the main tools for analyzing group structure.

Continue studying groups through the groups topic, then connect these ideas to rings, fields, number theory, permutation theory, symmetry, cosets, quotient groups, and group actions.