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The symmetric group Sₙ is the group of all bijections (permutations) of an n-element set under composition. Permutation groups are fundamental: every finite group is a subgroup of some symmetric group. The order of Sₙ is n!.
A permutation of a set is simply a rearrangement — a bijection from the set to itself. If your set is {1, 2, 3}, one permutation sends 1→2, 2→3, 3→1 (a cyclic rotation) and another sends 1→2, 2→1, 3→3 (a swap, called a transposition). From your study of the group axioms, you can verify that the set of all permutations of an n-element set forms a group under function composition: composing two bijections gives a bijection, the identity permutation acts as the identity element, and every bijection has an inverse (the reverse mapping). This group is the symmetric group Sₙ.
The order of Sₙ is n!, since there are n! ways to arrange n distinct objects. S₂ has just 2 elements; S₃ has 6; S₄ has 24. Even S₃ is already non-abelian: if σ rotates {1,2,3} cyclically and τ swaps 1 and 2, then σ∘τ and τ∘σ send elements to different places. This makes Sₙ the first natural source of non-abelian groups in abstract algebra. Every multiplication table you can draw for a 6-element non-abelian group will turn out to be the table of S₃ — it is the unique non-abelian group of order 6.
The symmetric group is important far beyond combinatorics. Cayley's theorem states that every finite group is isomorphic to a subgroup of Sₙ for some n. This means permutation groups are *universal*: any abstract finite group can be concretely realized as a group of rearrangements. Rather than reasoning about arbitrary groups in the abstract, you can always find a copy inside some Sₙ and compute by shuffling elements of a set. This is analogous to the way every finite-dimensional vector space can be realized as ℝⁿ — a concrete coordinate model always exists.
A key subgroup of Sₙ is the alternating group Aₙ, consisting of all "even" permutations — those expressible as a product of an even number of transpositions. Aₙ has order n!/2 and plays a central role in Galois theory: the fact that A₅ is simple (has no normal subgroups) is the reason no general formula exists for solving quintic equations. For now, the main skill to develop is fluency with permutation composition — writing permutations explicitly, composing them left-to-right or right-to-left consistently, and recognizing that the result depends on the order of composition.