Difference between revisions of "Free group"
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An example of an element of the free group on <math>I = \{1, 2\}</math> is <math>X_1X_2^{-1}X_1^{-1}X_2^3</math> (where by <math>X_2^3</math> we mean <math>X_2X_2X_2</math>). | An example of an element of the free group on <math>I = \{1, 2\}</math> is <math>X_1X_2^{-1}X_1^{-1}X_2^3</math> (where by <math>X_2^3</math> we mean <math>X_2X_2X_2</math>). | ||
− | The [[inverse with respect to an operation | inverse]] of a given element of a free group can be found by reversing the string and the sign of all exponents. Thus, the string given above has inverse <math>X_2^{-3}X_1X_2X_1^{-1}</math>. This is easy to check: we just apply the group operation to the two strings to get <math>X_2^{-3}X_1X_2X_1^{-1}X_1X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2\cdot1\cdot X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1\cdot 1\cdotX_1^{-1}X_2^3 = X_2^{-3}X_1X_1^{-1}X_2^3 = X_2^{-3}\cdot 1\cdotX_2^3 = X_2^{-3}X_2^3 = 1</math>. (Note that we implicitly used the [[associativity]] of the group operation repeatedly in this process.) Showing that this string is a right inverse is equally straightforward. The proof that this holds for every string is by [[induction]] using the same idea. | + | The [[inverse with respect to an operation | inverse]] of a given element of a free group can be found by reversing the string and the sign of all exponents. Thus, the string given above has inverse <math>X_2^{-3}X_1X_2X_1^{-1}</math>. This is easy to check: we just apply the group operation to the two strings to get <math>\displaystyle X_2^{-3}X_1X_2X_1^{-1}X_1X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2\cdot1\cdot X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1\cdot 1\cdotX_1^{-1}X_2^3 = X_2^{-3}X_1X_1^{-1}X_2^3 = X_2^{-3}\cdot 1\cdotX_2^3 = X_2^{-3}X_2^3 = 1</math>. (Note that we implicitly used the [[associativity]] of the group operation repeatedly in this process.) Showing that this string is a right inverse is equally straightforward. The proof that this holds for every string is by [[induction]] using the same idea. |
More formally, free groups are defined by [[universal property|universal properties]]. A group <math>F</math> is called a free group on <math>I</math> if there is a function <math>\phi:I\to F</math> so that for any group <math>G</math> and a function <math>\theta:I\to G</math>, there is a unique group homomorphism <math>\psi:F\to G</math> so that <math>\theta=\psi\phi</math>, i.e. so that <math>\theta(i)=\psi\circ\phi(i)</math> for all <math>i\in I</math>. We often like to draw a diagram to represent this relationship; however WE DON'T HAVE THE xy PACKAGE INCLUDED, SO I CAN'T TeX IT. | More formally, free groups are defined by [[universal property|universal properties]]. A group <math>F</math> is called a free group on <math>I</math> if there is a function <math>\phi:I\to F</math> so that for any group <math>G</math> and a function <math>\theta:I\to G</math>, there is a unique group homomorphism <math>\psi:F\to G</math> so that <math>\theta=\psi\phi</math>, i.e. so that <math>\theta(i)=\psi\circ\phi(i)</math> for all <math>i\in I</math>. We often like to draw a diagram to represent this relationship; however WE DON'T HAVE THE xy PACKAGE INCLUDED, SO I CAN'T TeX IT. |
Revision as of 09:25, 24 August 2006
A free group is a type of group that is of particular importance in combinatorics. Let be any nonempty index set. Informally, a free group on
is the collection of finite strings of characters from the collection
subject only to the criterion that
where
is the group identity and is equal to the empty string. The group operation is concatenation.
An example of an element of the free group on is
(where by
we mean
).
The inverse of a given element of a free group can be found by reversing the string and the sign of all exponents. Thus, the string given above has inverse . This is easy to check: we just apply the group operation to the two strings to get $\displaystyle X_2^{-3}X_1X_2X_1^{-1}X_1X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2\cdot1\cdot X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1X_2X_2^{-1}X_1^{-1}X_2^3 = X_2^{-3}X_1\cdot 1\cdotX_1^{-1}X_2^3 = X_2^{-3}X_1X_1^{-1}X_2^3 = X_2^{-3}\cdot 1\cdotX_2^3 = X_2^{-3}X_2^3 = 1$ (Error compiling LaTeX. Unknown error_msg). (Note that we implicitly used the associativity of the group operation repeatedly in this process.) Showing that this string is a right inverse is equally straightforward. The proof that this holds for every string is by induction using the same idea.
More formally, free groups are defined by universal properties. A group is called a free group on
if there is a function
so that for any group
and a function
, there is a unique group homomorphism
so that
, i.e. so that
for all
. We often like to draw a diagram to represent this relationship; however WE DON'T HAVE THE xy PACKAGE INCLUDED, SO I CAN'T TeX IT.
Commutivity in Free Groups
The free group over has "as little structure as possible." Thus, we can show that two elements of the free group commute only when it's "necessary":
If
are elements of the free group over a set
and
, then
or
.
Proof
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