Probability of satisfaction of a collection of words

Definition

For a finite group and a single word

Suppose $G$ is a finite group and $w$ is a word in $n$ letters. $w$ defines a word map from $G^{n}$ to $G$ by sending any $n$ -tuple in $G$ to the word $w$ evaluated at that $n$ -tuple. The probability of satisfaction of $w$ is the probability (under the uniform distribution on $G^{n}$ ) that the image of the word map gives the identity element of $G$ .

Explicitly, denote by $S_{w}$ the subset of $G^{n}$ given by:

$S_{w} = {(g_{1}, g_{2}, \dots, g_{n}) ∣ w (g_{1}, g_{2}, \dots, g_{n}) = e (the identity element of G)}$

Then, the probability of satisfaction of $w$ is the quotient:

$\frac{| S_{w} |}{| G |^{n}}$

For a finite group and a collection of words

Suppose $G$ is a finite group and $C$ is a collection of words, all of which use a fixed alphabet of $n$ letters. Thus, each $w \in C$ defines a word map from $G^{n}$ to $G$ by sending any $n$ -tuple in $G$ to the word $w$ evaluated at that $n$ -tuple. The probability of satisfaction of $C$ is the probability (under the uniform distribution on $G^{n}$ ) that the image of the word map for every $w \in C$ gives the identity element of $G$ .

Explicitly, denote by $S_{C}$ the subset of $G^{n}$ given by:

$S_{C} = {(g_{1}, g_{2}, \dots, g_{n}) ∣ w (g_{1}, g_{2}, \dots, g_{n}) = e (the identity element of G) \forall w \in C}$

Then, the probability of satisfaction of $C$ is the quotient:

$\frac{| S_{C} |}{| G |^{n}}$

Facts

Facts for probability of satisfaction applicable to single words and to collections of words

The probability of satisfaction of $w$ in $G$ is at least $1 / | G |^{n}$ where $n$ is the number of letters appearing in $w$ . This is because the tuple with all coordinates the identity element must satisfy the word.
Probability of satisfaction is invariant under Andrews-Curtis transformations: The probability of satisfaction of a word is invariant under conjugation. It is also invariant under performing Nielsen transformations on the word in the free group on $n$ letters. Thus, it is invariant under Andrews-Curtis transformations. The same result holds for a collection of words, but we need to do the Andrews-Curtis transformations on the subset together rather than doing a different transformation on each word.
The probability of satisfaction of $w$ in $G$ is 1 if and only if $G$ is in the subvariety of the variety of groups where $w$ is satisfied (i.e., where $w$ equals the identity element). Similarly, the probability of satisfaction of a collection $C$ of words in $G$ is 1 if and only if $G$ is in the subvariety of the variety of groups where all $w \in C$ are satisfied.
Words defining the same variety may have different probability of satisfaction: Note that even if two words define the same subvariety of the variety of groups, the probability of satisfaction of the two words may be different. For instance, the word $x^{2}$ (in one letter) and the word $x^{2} y^{2}$ (in two letters) define the same subvariety -- the variety of elementary abelian 2-groups. However, in the quaternion group, the former word has probability of satisfaction 1/4 whereas the latter word has probability of satisfaction 5/8.

Facts for probability of satisfaction applicable only to single words

If there exists a letter in the word that appears just once in the word with an exponent of 1 or -1, then the probability of satisfaction of the word is $1 / | G |$ . The reason is that for any choice of values of the other letters, there will always be a unique choice of that letter for which the word is the identity element. Thus, for instance, the words $x y, x y^{- 1}, x^{2} y^{- 1}, x y^{- 1} x, x y x^{- 1}, y^{2} x z^{- 3}$ , all have probability of satisfaction $1 / | G |$ .

Particular cases

Word	Number of letters	Probability of satisfaction	Explanation
$x$	1	$1 / \| G \|$	true if and only if the chosen element is the identity element.
$x^{2}$	1	(1 + number of involutions in $G$ )/ $\| G \|$	true if the element is either the identity element or has order two
$x y x^{- 1} y^{- 1}$	2	commuting fraction (probability that two elements commute)	true if and only if the elements commute
$[[x, y], z]$ ( $[,]$ denotes the commutator, this can be expanded to a word)	3	class two fraction	by definition of class two fraction

References

MathOverflow question (not completely resolved) on probability of satisfaction of a word