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Non-Abelian — Definition, Formula & Examples

Non-Abelian is a term describing a group in which the order you combine two elements can change the result. In other words, there exist elements aa and bb in the group such that a∗b≠b∗aa * b \neq b * a.

A group (G,∗)(G, *) is non-Abelian if its binary operation is not commutative — that is, if there exist at least two elements a,b∈Ga, b \in G such that a∗b≠b∗aa * b \neq b * a. This stands in contrast to an Abelian group, where a∗b=b∗aa * b = b * a for all a,b∈Ga, b \in G.

How It Works

To determine whether a group is non-Abelian, you only need to find one pair of elements whose product depends on the order. If every possible pair commutes, the group is Abelian. The smallest non-Abelian group is S3S_3, the symmetric group on three elements, which has order 6. Matrix multiplication under GLn(R)GL_n(\mathbb{R}) is another classic source of non-Abelian behavior, since multiplying matrices in different orders typically yields different results.

Worked Example

Problem: Show that the symmetric group S3S_3 is non-Abelian by finding two permutations that do not commute.
Step 1: Define two permutations in S3S_3. Let σ=(1 2)\sigma = (1\ 2) (swap 1 and 2) and τ=(1 2 3)\tau = (1\ 2\ 3) (cycle 1→2→3→1).
σ=(1 2),τ=(1 2 3)\sigma = (1\ 2), \quad \tau = (1\ 2\ 3)
Step 2: Compute σ∘τ\sigma \circ \tau. Apply τ\tau first, then σ\sigma: 1→τ2→σ11 \xrightarrow{\tau} 2 \xrightarrow{\sigma} 1, 2→τ3→σ32 \xrightarrow{\tau} 3 \xrightarrow{\sigma} 3, 3→τ1→σ23 \xrightarrow{\tau} 1 \xrightarrow{\sigma} 2.
σ∘τ=(2 3)\sigma \circ \tau = (2\ 3)
Step 3: Compute τ∘σ\tau \circ \sigma. Apply σ\sigma first, then τ\tau: 1→σ2→τ31 \xrightarrow{\sigma} 2 \xrightarrow{\tau} 3, 2→σ1→τ22 \xrightarrow{\sigma} 1 \xrightarrow{\tau} 2, 3→σ3→τ13 \xrightarrow{\sigma} 3 \xrightarrow{\tau} 1.
τ∘σ=(1 3)\tau \circ \sigma = (1\ 3)
Step 4: Compare the results. Since (2 3)≠(1 3)(2\ 3) \neq (1\ 3), the two compositions differ.
σ∘τ≠τ∘σ\sigma \circ \tau \neq \tau \circ \sigma
Answer: Since σ∘τ≠τ∘σ\sigma \circ \tau \neq \tau \circ \sigma, the group S3S_3 is non-Abelian.

Why It Matters

Non-Abelian groups are central to advanced algebra and physics. The classification of finite simple groups depends heavily on non-Abelian structure, and gauge theories in particle physics (like the Standard Model) are built on non-Abelian symmetry groups such as SU(3)SU(3) and SU(2)SU(2).

Common Mistakes

Mistake: Assuming a group is non-Abelian because one specific pair of elements does not commute, without verifying both elements are actually in the group.
Correction: Always confirm the elements belong to the group in question. Conversely, to prove a group is Abelian, you must show commutativity holds for all pairs, not just a few.