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

Example

Let $\mathbb Z/2$ act on the singleton $X=\{*\}$ by $a\cdot *=*$ for both residue classes $a$ . This action is transitive, but it is not faithful.

Facts & Assumptions

Given: The additive group $\mathbb Z/2$ and the singleton set $X=\{*\}$ .

[L1]

Division with remainder is available in the integers, and congruence classes modulo $2$ are the quotient group with its stated addition and identity class $0$ (Division with remainder in $\mathbb{Z}$ : for $a \in \mathbb{Z}$ and $b > 0$ there are unique $q, r \in \mathbb{Z}$ with $a = qb + r$ and $0 \le r < b$ , For every $n\in\mathbb N$ , the congruence-class group $(\mathbb Z/n,+)$ is the quotient group $(\mathbb Z,+)/n\mathbb Z$ , For every natural $n$ , $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold, Addition and multiplication on $\mathbb{Z}/n$ by $[a]_n+[b]_n=[a+b]_n$ and $[a]_n[b]_n=[ab]_n$ ).

[L2]

An action satisfies $0\cdot x=x$ and $(a+b)\cdot x=a\cdot(b\cdot x)$ ; it is transitive when one group element carries every point to every other point, and faithful only when an element fixing every point is the identity (Left group actions, transitive actions, and faithful actions).

Verification

technique · direct

1.1

Dividing any integer by $2$ shows that the two residue classes are $0$ and $1$ ; they are distinct because $1-0$ is not a multiple of $2$ , and $1+1=0$ .

L1algebra

1.2

The rule $a\cdot *=*$ satisfies $0\cdot *=*$ and $(a+b)\cdot *=*=a\cdot(b\cdot *)$ , so it is an action.

L1L2given

1.3

There is only one point of $X$ , so the action is transitive.

L2given

2.1

The nonidentity class $1$ fixes $*$ , and therefore fixes every point of $X$ . Hence the action is not faithful.

step 1.1L2given∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Brosnan, Group actions (standard reference, not scraped)

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