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Could you help me to find a set of normal forms for the group $\langle a,b,c| a^2=b^2=c^2=1, (ab)^2=(ac)^4=1\rangle$?

EDIT: how can I draw a cayley graph for this group?

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$a$ and $c$ generate a dihedral group of order $8$; $a$ and $b$ generate a Klein $4$-group. $b$ and $c$ generate a $C_2*C_2$. –  Arturo Magidin Mar 22 '12 at 5:22
    
In particular, this is an amalgamated product of $D_4$ and $D_8$, with some reflection ($a$) amalgamated. –  user641 Mar 22 '12 at 6:04

1 Answer 1

A normal form consists of all words in $a,b,c$ that contain none of the words $aa$, $bb$, $cc$, $ba$, $caca$ as subwords.

(This is because the set of rules

$aa \to \epsilon$, $bb \to \epsilon$, $cc \to \epsilon$, $ba \to ab$, $caca \to acac$

forms a confluent rewriting system for the group.)

Added later: There is a standard algorithm (generally known as Knuth-Bendix) for checking a rewriting system for completeness. You look for common prefixes and suffixes of left hand sides of rules. Each of these gives a word that can be reduced in two different ways, and you need to check that both reductions result in the same irreducible word. This is easy to do in this example, because there are not many common prefixes and suffixes.

One such is the prefix $a$ of $aa$ with the suffix $c$ of $caca$, allowing two different reductions of $cacaa$:

$cac(aa) \to cac$, and $(caca)a \to a(caca) \to (aa)cac \to cac$,

so both reductions result in $cac$.

Of course, for more complicated examples, this is best done by computer.

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Is it clear that you end up with a finite set? –  lhf Mar 22 '12 at 10:58
    
I have appended a method of checking to my post. –  Derek Holt Mar 22 '12 at 11:35
    
how can I do if I want to draw the cayley graph? –  John Mar 26 '12 at 23:30
    
@John: It is an infinite group, so drawing the Cayley graph might take a while... –  user1729 Mar 27 '12 at 14:56

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