free group
Noun
Meaning
free group (plural free groups)
(group theory) A group that has a presentation without relators; equivalently, a free product of some number of copies of ℤ.
Given a set S of "free generators" of a free group, let
S
−
1
{\displaystyle S^{-1}}
be the set of inverses of the generators, which are in one-to-one correspondence with the generators (the two sets are disjoint), then let
(
S
∪
S
−
1
)
∗
{\displaystyle (S\cup S^{-1})^{*}}
be the Kleene closure of the union of those two sets. For any string w in the Kleene closure let r(w) be its reduced form, obtained by cutting out any occurrences of the form
x
x
−
1
{\displaystyle xx^{-1}}
or
x
−
1
x
{\displaystyle x^{-1}x}
where
x
∈
S
{\displaystyle x\in S}
. Noting that r(r(w)) = r(w) for any string w, define an equivalence relation
∼
{\displaystyle \sim }
such that
u
∼
v
{\displaystyle u\sim v}
if and only if
r
(
u
)
=
r
(
v
)
{\displaystyle r(u)=r(v)}
. Then let the underlying set of the free group generated by S be the quotient set
(
S
∪
S
−
1
)
∗
/
∼
{\displaystyle (S\cup S^{-1})^{*}/\sim }
and let its operator be concatenation followed by reduction.
Examples
One was one of their ex-party members and the voice and scribe behind a lot of the Keep Bodden Town Dump Free group.