Alphabeta Math

Group Actions: Examples and Counterexamples

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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

The trivial action of ℤ/2 on a singleton is transitive but not faithful

Example

Left multiplication gives a free and transitive action of every group on itself

Counterexample

The natural action of S₃ on three points is faithful and transitive but not free

Counterexample

The action of ℤ/2 on two disjoint two-point orbits is free but not transitive

Example

The four rotations of a square act freely, transitively and faithfully on its vertices

Example

The action of ℤ/6 on the cosets of {0,3} is transitive with kernel {0,3} and is not faithful

Example

The class equation of S₃ is 6=1+2+3

Counterexample

S₃ has trivial center, so the finite p-group hypothesis in the nontrivial-center theorem is necessary

Example

The square-symmetry group has class equation 8=2+2+2+2

Counterexample

The square-symmetry group has order 2³ and is nonabelian, so the order-p² theorem does not extend to order p³

Example

The three subgroups of order 2 in S₃ are conjugate and each is self-normalizing

Example

An involution on five points has three fixed points and one two-point orbit, verifying 5≡3 (mod 2)

Counterexample

S₃ acting on three points has |X|=3 and |X^S₃|=0, so the fixed-point congruence modulo 2 fails without the p-group hypothesis

Counterexample

The subgroup ⟨(1 2 3),(1 2)(3 4)⟩≤ S₄ has order 12 but no subgroup of order 6, so Cauchy's theorem does not extend to composite divisors

Example

There are six two-colourings of the vertices of a square up to its eight symmetries

Example

There are six binary necklaces of length four up to rotation

Counterexample

A nonfree action can have |X/G|≠|X|/|G|

Sources