Permutation, Symmetric, Alternating and Dihedral Groups
Permutation, Symmetric, Alternating and Dihedral Groups: core definitions, structural results and verification methods in abstract algebra.
How the topic fits together
Permutation Groups
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definition
A permutation of a set S is a bijection on S, that is, a function π : S →S that is one-to-one and onto. (If S is finite, then π is one-to-one if and only if it is onto.) If S is not too large, it is feasible to describe a permutation by listing the elements x ∈S and the corresponding values π(x). For example, if S = {1, 2, 3, 4, 5}, then π = 1 2 3 4 5 3 5 4 1 2 is the permutation such that π(1) = 3, π(2) = 5, π(3) = 4, π(4) = 1, π(5) = 2. If we start with any element x ∈S and apply π repeatedly to obtain π(x), π(π(x)), π(π(π(x))), and so on, eventually we must return to x, and there are no repetitions along the way because π is one-to-one.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definitions and Comments
A permutation π is called even if its cycle decomposition contains an even number of even cycles (that is, cycles of even length); otherwise π is odd. A cycle can be decomposed further into a product of (not necessarily disjoint) twoelement cycles, called transpositions. For example, (1, 2, 3, 4, 5) = (1, 5)(1, 4)(1, 3)(1, 2) where the order of application of the mappings is from right to left. Multiplication by a transposition changes the parity of a permutation (from even to odd, or vice versa).
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definitions and Comments
There are several permutation groups that are of major interest. The set Sn of all permutations of {1, 2, . . . , n} is called the symmetric group on n letters, and its subgroup An of all even permutations of {1, 2, . . . , n} is called the alternating group on n letters. (The group operation is composition of functions.) Since there are as many even permutations as odd ones (any transposition, when applied to the members of Sn, produces a one-to-one correspondence between even and odd permutations), it follows that An is half the size of Sn. Denoting the size of the set S by |S|, one has |Sn| = n!, |An| = 1 2n!
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
An Abstract Characterization of the Dihedral Group
An Abstract Characterization of the Dihedral Group Consider the free group with generators R and F, in other words all finite sequences whose components are R, R−1, F and F −1. The group operation is concatenation, subject to the constraint that if a symbol and its inverse occur consecutively, they may be cancelled. For example, RFFFF −1RFR−1RFF is identified with RFFRFFF, also written as RF 2RF 3. If we add further restrictions (so the group is no longer “free”), one can obtain D2n.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test order, subgroup, exact, prime, Tor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 1.2 | Permutation Groups | 12–14 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.