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Topic: abelian square
Replies: 5   Last Post: Sep 8, 2013 2:50 PM

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Peter Percival

Posts: 2,623
Registered: 10/25/10
Re: abelian square
Posted: Sep 8, 2013 8:58 AM
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Dieter von Holten wrote:
> hi folks,
> i found 'abelian squares' in the context of patterns of n symbols of length 2n like 'abcd.cdba' where 'cdba' is a permutation of the first half 'abcd'.
> however, in those texts it is not explained (maybe just not clear enough for me) what is 'abelian' and where is the 'square' ??
> any hints ?

I don't know so I'm going to guess. (Which means if the answer is
important to you, you should disregard this post.)

Let a,b,c,...,a^{-1},b^{-1},c^{-1},... be an alphabet of symbols. Let
"words" be concatenations of those symbols, the empty word is included,
and the concatenation of x and x^{-1}, and x^{-1} and x, is the empty
word. Then the set of words is a group under concatenation. If, for
words U and V, the concatenation UV is deemed equal to the concatenation
VU, the the group is abelian. The concatenation of U with itself, UU,
might reasonable be called a square.

So, in such an abelian group, abcd.cdba = abcd.abcd, a square. Here .
denotes concatenation, and the identity is demonstrated by applying
association and commutativity repeatedly.

Note, monoids could be considered in place of groups, but am I right in
my belief that "abelian" is not often used of monoids, instead one says

[Warning: The operation that carries abcc^{-1}de to abde (for example)
is called a reduction and abde is called a reduced word (assuming
distinct letters are indeed distinct alphabet symbols). The group of
reduced words is called the free group on the alphabet, _but_ such a
free group that is abelian is generally not the same as the thing called
a free abelian group. "Generally" because the trivial group and the
infinite cyclic group (=(Z,+)) are, indeed, both abelian free groups and
free abelian groups. They are the only groups for which the concepts

Sorrow in all lands, and grievous omens.
Great anger in the dragon of the hills,
And silent now the earth's green oracles
That will not speak again of innocence.
David Sutton -- Geomancies

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