Word Problem For Groups - Algebraic Structure and The Word Problem

Algebraic Structure and The Word Problem

There are a number of results that relate solvability of the word problem and algebraic structure. The most significant of these is the Boone-Higman theorem:

A finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group.

It is widely believed that it should be possible to do the construction so that the simple group itself is finitely presented. If so one would expect it to be difficult to prove as the mapping from presentations to simple groups would have to be non-recursive.

The following has been proved by Bernhard Neumann and Angus Macintyre:

A finitely presented group has solvable word problem if and only if it can be embedded in every algebraically closed group

What is remarkable about this is that the algebraically closed groups are so wild that none of them has a recursive presentation.

The oldest result relating algebraic structure to solvability of the word problem is Kuznetsov's theorem:

A recursively presented simple group S has solvable word problem.

To prove this let ⟨X|R⟩ be a recursive presentation for S. Choose a ∈ S such that a ≠ 1 in S.

If w is a word on the generators X of S, then let:

There is a recursive function such that:

f_{\langle X | R\cup \{w\} \rangle}(x) =
\left\{\begin{matrix}
0 &\mbox{if}\ x=1\ \mbox{in}\ S_w\\
\mbox{undefined/does not halt}\ &\mbox{if}\ x\neq 1\ \mbox{in}\ S_w.
\end{matrix}\right.

Write:

Then because the construction of f was uniform, this is a recursive function of two variables.

It follows that: h(w)=g(w, a) is recursive. By construction:

h(w) =
\left\{\begin{matrix}
0 &\mbox{if}\ a=1\ \mbox{in}\ S_w\\
\mbox{undefined/does not halt}\ &\mbox{if}\ a\neq 1\ \mbox{in}\ S_w.
\end{matrix}\right.

Since S is a simple group, its only quotient groups are itself and the trivial group. Since a ≠ 1 in S, we see a = 1 in Sw if and only if Sw is trivial if and only if w ≠ 1 in S. Therefore:

h(w) =
\left\{\begin{matrix}
0 &\mbox{if}\ w\ne 1\ \mbox{in}\ S\\
\mbox{undefined/does not halt}\ &\mbox{if}\ w=1\ \mbox{in}\ S.
\end{matrix}\right.

The existence of such a function is sufficient to prove the word problem is solvable for S.

This proof does not prove the existence of a uniform algorithm for solving the word problem for this class of groups. The non-uniformity resides in choosing a non-trivial element of the simple group. There is no reason to suppose that there is a recursive function that maps a presentation of a simple groups to a non-trivial element of the group. However, in the case of a finitely presented group we know that not all the generators can be trivial (Any individual generator could be, of course). Using this fact it is possible to modify the proof to show:

The word problem is uniformly solvable for the class of finitely presented simple groups.

Read more about this topic:  Word Problem For Groups

Famous quotes containing the words algebraic, structure, word and/or problem:

    I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
    Henry David Thoreau (1817–1862)

    Agnosticism is a perfectly respectable and tenable philosophical position; it is not dogmatic and makes no pronouncements about the ultimate truths of the universe. It remains open to evidence and persuasion; lacking faith, it nevertheless does not deride faith. Atheism, on the other hand, is as unyielding and dogmatic about religious belief as true believers are about heathens. It tries to use reason to demolish a structure that is not built upon reason.
    Sydney J. Harris (1917–1986)

    The word of the Lord by night
    To the watching Pilgrims came,
    As they sat by the seaside,
    And filled their hearts with flame.
    Ralph Waldo Emerson (1803–1882)

    The problem is simply this: no one can feel like CEO of his or her life in the presence of the people who toilet trained her and spanked him when he was naughty. We may have become Masters of the Universe, accustomed to giving life and taking it away, casually ordering people into battle or out of their jobs . . . and yet we may still dirty our diapers at the sound of our mommy’s whimper or our daddy’s growl.
    Frank Pittman (20th century)