Groups: Axioms and First Examples

Part 11, Chapter 11: Abstract Algebra Survey

Learning objectives

State the four group axioms (closure, associativity, identity, inverses) and the extra abelian axiom
Recognise standard examples: $(\mathbb{Z}, +)$ , $\mathbb{Z}/n\mathbb{Z}$ , $S_n$ , $GL_n(\mathbb{R})$ , $D_n$
Identify the order of an element and the order of a group, and apply Lagrange's theorem to constrain subgroup sizes
Distinguish abelian from non-abelian groups using concrete computations in $S_3$

A group is the minimum amount of structure you need to talk about symmetry. Strip away everything that distinguishes integers, permutations, rotations, and matrices and what is left, one binary operation, an identity, and inverses, is a group. The discovery that this thin axiom system captures the symmetry of geometry, the solubility of polynomial equations, the conservation laws of physics, and the structure of error-correcting codes is one of the great unifications in mathematics.

The four axioms

A group is a pair $(G, \cdot)$ where $G$ is a set and $\cdot : G \times G \to G$ is a binary operation satisfying:

Closure: for all $a, b \in G$ , the product $a \cdot b$ lies in $G$ . (Often folded into the definition of "binary operation," but worth stating once.)
Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for every $a, b, c \in G$ .
Identity: there exists $e \in G$ with $e \cdot a = a \cdot e = a$ for every $a$ .
Inverses: for every $a \in G$ there exists $a^{-1} \in G$ with $a \cdot a^{-1} = a^{-1} \cdot a = e$ .

If additionally $a \cdot b = b \cdot a$ for all elements, the group is abelian. The integers under addition are abelian; the symmetric group $S_3$ is not.

Four examples to keep in your head

The integers under addition $(\mathbb{Z}, +)$ : identity $0$ , inverse of $n$ is $-n$ , infinite and abelian. The cyclic group $\mathbb{Z}/n\mathbb{Z} = \{0, 1, \ldots, n-1\}$ under addition mod $n$ : a finite abelian group of order $n$ , the simplest finite group to compute with. The symmetric group $S_n$ n of all permutations of $\{1, \ldots, n\}$ under composition: order $n!$ , non-abelian for $n \geq 3$ , and the universal home for every finite group via Cayley's theorem. The general linear group $GL_n(\mathbb{R})$ n(mathbbR) of invertible $n \times n$ real matrices under multiplication: continuous, non-abelian for $n \geq 2$ , and the natural setting for representation theory.

Order of an element, order of a group

The order of a group is its cardinality $|G|$ . The order of an element $a$ is the smallest positive integer $k$ with $a^k = e$ (or $\infty$ if no such $k$ exists). In $\mathbb{Z}/12\mathbb{Z}$ the element $4$ has order $3$ : $4, 8, 0$ . Lagrange's theorem says the order of any subgroup divides $|G|$ ; in particular the order of any element divides $|G|$ . A group of order $7$ has only orders $1$ and $7$ available for its elements, so every non-identity element generates the whole group, and $\mathbb{Z}/7\mathbb{Z}$ is the unique group of order $7$ up to isomorphism.

The clock above is a picture of $\mathbb{Z}/n\mathbb{Z}$ as a group: the hand position is the element, the operation "add $k$ " is "step $k$ marks clockwise," and the identity is the $0$ mark. Try $n = 12$ and watch how adding $5$ four times lands on $8$ , that is $4 \cdot 5 \equiv 8 \pmod{12}$ , the additive group law made geometric.

Where this shows up

Particle physics: The Standard Model is built on the gauge group $SU(3) \times SU(2) \times U(1)$ , three matrix groups whose representations classify quarks, leptons, gluons, and the electroweak bosons. Conservation of charge, colour, and weak isospin are direct consequences of group invariance.
Molecular chemistry: Every molecule has a point group describing its rotational and reflective symmetries. The character table of that group determines which vibrational modes are infrared-active, which electronic transitions are allowed, and which orbitals can mix. Spectroscopy is applied finite group theory.
Cryptography: RSA, Diffie-Hellman, and elliptic-curve cryptography all run inside groups, the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^\times$ for RSA, the additive group of points on an elliptic curve for ECC. Security reduces to "the discrete-log problem is hard in this group."
Rubik's Cube: The set of legal cube positions is a group of order $\approx 4.3 \times 10^{19}$ . Every published solving algorithm is a sequence of generators, and "God's number" (20 moves) was computed by exhaustively searching the cosets of a strategic subgroup.
Cellular networks: The handover protocols in 5G rely on group-theoretic codes (Reed-Muller, polar codes) where the symmetry group acts on codewords to give predictable decoding behaviour.

Pause and think: Is $(\mathbb{Z}, \times)$ a group under ordinary multiplication? Walk through the four axioms one at a time. (Hint: which axiom fails, closure, associativity, identity, or inverses?)

Try it

Predict first: in $\mathbb{Z}/6\mathbb{Z}$ , what is the order of the element $4$ ? Compute $4, 4+4, 4+4+4, \ldots$ mod $6$ and find the first return to $0$ .
Write out the multiplication table for $\mathbb{Z}/3\mathbb{Z} = \{0, 1, 2\}$ under addition mod $3$ . Verify each row and column is a permutation of $\{0, 1, 2\}$ , this is the Latin square property of group tables.
Find two permutations $\sigma, \tau \in S_3$ with $\sigma \tau \neq \tau \sigma$ . Try $\sigma = (1\;2)$ and $\tau = (2\;3)$ as transpositions and compute both compositions.
Verify that the set of positive rationals $\mathbb{Q}_{>0}$ >0 under multiplication IS a group, then state its identity element and the inverse of a generic element $a/b$ . State the identity and the inverse of $a/b$ .
Lagrange in action: a group has order $15$ . What are the possible orders of its elements? (Use that the order of an element divides the order of the group.)

A trap to watch for

The most common beginner error is forgetting that the operation is part of the data, $(\mathbb{Z}, +)$ is a group, but $(\mathbb{Z}, \times)$ is not: $2$ has no multiplicative inverse inside $\mathbb{Z}$ . Worse, the same set can be a group under one operation and not under another. A second common trap: assuming all groups are commutative. Once $|G| \geq 6$ non-abelian groups appear (the symmetric group $S_3$ and the dihedral group $D_3$ both have order $6$ ), and you cannot rearrange products freely. Always check, explicitly, whether you are inside an abelian group before swapping factor order.

What you now know

You can state the four group axioms, classify whether a given $(S, \cdot)$ is a group, compute orders of elements in finite cyclic and symmetric groups, and apply Lagrange to rule out impossible subgroup sizes. The next section turns this around: instead of studying groups as abstract objects, we represent them by linear maps so the whole machinery of linear algebra becomes available.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 11.

Dummit, D. S., Foote, R. M. (2003). Abstract Algebra (3rd ed.). Wiley, ch. 1-3.

Artin, M. (2010). Algebra (2nd ed.). Pearson, ch. 2.

Lang, S. (2002). Algebra (3rd revised ed.). Springer, ch. 1.

Herstein, I. N. (1996). Abstract Algebra (3rd ed.). Wiley, ch. 2.

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