Difference between revisions of "Commutator (group)"

Latest revision as of 23:55, 1 June 2008

This article refers to commutators of groups, not to be confused with commutator groups.

In a group, the commutator of two elements $a$ and $b$ , denoted $(a,b)$ or $[a,b]$ , is the element $a^{-1}b^{-1}ab$ . If $a$ and $b$ commute, then $(a,b)=e$ . More generally, $(a,b) = (ba)^{-1}ab$ , or $ab = ba(a,b) .$ It then follows that $(a,b)(b,a) = e .$ We also have $x^y = y^{-1}xy = x(x,y) = (y,x^{-1})x ,$ where $x^y$ denote the image of $x$ under the inner automorphism $\text{Int}(y^{-1})$ , as usual.

Relations with Commutators

Proposition 1. For all $x,y,z$ in a group, the following relations hold:

$(x,yz) = (x,z)(x,y)^z = (x,z)(z,(y,x))(x,y)$ ;
$(xy,z) = (x,z)^y (y,z) = (x,z)((x,z),y)(y,z)$ ;
$(x^y,(y,z))(y^z,(z,x))(z^x,(x,y)) = e$ ;
$(x,yz)(z,xy)(y,zx) = e$ ;
$(xy,z)(yz,x)(zx,y) = e$ .

Proof. For the first equation, we note that $(x,yz) = x^{-1}z^{-1}y^{-1}xyz = (x^{-1}z^{-1}xz)z^{-1}(x^{-1}y^{-1}xy)z = (x,z) (x,y)^z .$ From the earlier relations, $(x,y)^z = (z,(x,y)^{-1})(x,y) = (z,(y,x))(x,y) ,$ hence the relation. The second equation follows from the first by passing to inverses.

For the third equation, we define $f(a,b,c) = ca (b)^c$ . We then note that $\begin{align*} (x^y,(y,z)) &= (x^{-1})^y \cdot (z,y) \cdot x^y \cdot (y,z) \\ &= (x^{-1})^y \cdot z^{-1} (z)^y \cdot x^y \cdot (z^{-1})^y \cdot z &= (x^{-1})^y \cdot z^{-1} \cdot y^{-1}(zxz^{-1})y \cdot z \\ &= \bigl[(x^{-1})^y \cdot z^{-1}y^{-1}\bigr] zx \cdot y^z \\ &= f(z,x,y)^{-1} f(x,y,z) . \end{align*}$ By cyclic permutation of variables, we thus find $\begin{align*} (x^y, (y,z))(y^z, (z,x))(z^x, (x,y)) &= f(z,x,y)^{-1} f(x,y,z) f(x,y,z)^{-1} f(y,z,x) f(y,z,x)^{-1} f(z,x,y) \\ &= e. \end{align*}$

For the fourth equation, we have $(x,yz)(z,xy)(y,zx) = (yzx)^{-1}(xyz)(xyz)^{-1} (zxy) (zxy)^{-1}(yzx) = e .$ The fifth follows similarly. $\blacksquare$

Commutators and Subgroups

If $A$ and $B$ are subgroups of a group $G$ , $(A,B)$ denotes the subgroup generated by the set of commutators of the form $(a,b)$ , for $a\in A$ and $b\in B$ .

The group $(A,B)$ is trivial if and only if $A$ centralizes $B$ . Also, $(A,B) \subseteq A$ if and only if $B$ normalizes $A$ . If $A$ and $B$ are both normal (or characteristic), then so is $(A,B)$ , for if $f$ is an (inner) automorphism, then $f((a,b)) = (f(a),f(b)).$ Note also that since $(a,b) = (b,a)^{-1}$ , $(A,B)=(B,A)$ .

Lemma. Let $A,B$ be a closed subsets of $G$ (not necessarily a subgroups); denote by $(A,B)$ the subgroup of $G$ generated by elements of the form $(a,b)$ , for $a\in A$ , $b\in B$ . Then $(A,B)^a \subseteq$ .

Proof. Let $a,a'$ be elements of $A$ and $b$ be an element of $B$ . Then $(a,b)^{a'} = a'^{-1}a^{-1}b^{-1}aba' = (aa',b)b^{-1}a'^{-1}aba'$

Proposition 2. Let $A,B,C$ be three subgroups of $G$ .

The group $A$ normalizes the group $(A,B)$ .
If the group $(B,C)$ normalizes $A$ , then the set of elements $(a,(b,c))$ , for $a\in A$ , $b\in B$ , $c \in C$ , generates the group $(A,(B,C))$ .
If $A$ , $B$ , and $C$ are normal subgroups of $G$ , then $(A,(B,C))$ is a subgroup of the group $(C,(B,A)) \cdot (B, (C,A))$ .

Proof. For the first, we note that for any $a,a' \in A$ and any $b \in B$ , $(a,b)^{a'} = (aa',b) (a',b)^{-1},$ by Proposition 1.

For the second, we have for any $x\in G$ , $a\in A$ , $b\in B$ , $c \in C$ , $\begin{align*} (a,(b,c)x) &= a^{-1}x^{-1}(c,b)a(b,c)x = (a,x)x^{-1}a^{-1}(c,b)a(b,c)x \\ &= (a,x) (x,((b,c),a)) (a,(b,c)) . \end{align*}$ Since $(B,C)$ normalizes $A$ , the element $((b,c),a)$ lies in $A$ . It then follows from induction on $n$ that for all $b_i \in B$ , $c_i \in C$ , the element $\biggl( a, \prod_{i=1}^n (b_i,c_i) \biggr)$ lies in the subgroup generated by elements of the form $(a,(b,c))$ . Similarly, $\begin{align*} (a,(b,c)) &= (a, (c,b) \cdot (b,c)^2) \\ &= (a,(b,c)^2)\Bigl( (b,c)^2,\bigl((c,b),a \bigr) \Bigr) \cdot (a,(c,b)) , \end{align*}$ lies in the subgroup generated by elements of the form $(a,(b,c))$ ; it then follows that $(a,(c,b))$ does. Then using the observation $(a,(c,b)x) = (a,x)(x, ((c,b),a)) (a,(c,b)),$ we prove by induction on $n$ that the element $\biggl(a, \prod_{i=1}^n (b_i,c_i)^{\pm 1} \biggr)$ lies in the subgroup generated by elements of the form $(a,(b,c))$ . This proves the second result.

For the third part, we first note that since $A,B,C$ are normal subgroups, so are $(A,(B,C))$ , $(C,(B,A))$ , and $(B,(C,A))$ ; in particular, $(C,(B,A)) \cdot (B,(C,A)) = (B,(C,A)) \cdot (C,(B,A))$ is a group. From the previous result of this proposition, it suffices to show that $(a,(b,c))$ lies in this group, for all $a\in A$ , $b\in B$ , and $c\in C$ . To that end, we note that since $A$ is normal, there exists $a' \in A$ such that $a = a'^{b}$ . Then by the third result of Proposition 1, $(a'^b , (b,c)) \cdot (b^c , (c,a')) \cdot (c^{a'} , (a',b)) = e,$ or

\[(a,(b,c)) = \bigl[ (b^c, (c,a')) \cdot (c^{a'}} , (a',b)) \bigr]^{-1} \in (B,(C,A)) \cdot (C, (B,A)) ,\] (Error compiling LaTeX. Unknown error_msg)

as desired. $\blacksquare$

@@ Line 1: / Line 1: @@
+''This article refers to [[commutator]]s of [[group]]s, not to be confused with [[commutator group]]s.''
 In a [[group]], the '''[[commutator]]''' of two elements <math>a</math> and <math>b</math>, denoted <math>(a,b)</math> or <math>[a,b]</math>, is the element <math>a^{-1}b^{-1}ab</math>.  If <math>a</math> and <math>b</math> commute, then <math>(a,b)=e</math>.  More generally, <math>(a,b) = (ba)^{-1}ab</math>, or
 <cmath> ab = ba(a,b) . </cmath>
@@ Line 9: / Line 11: @@
 == Relations with Commutators ==
-'''Proposition.''' For all <math>x,y,z</math> in a group, the following relations hold:
+'''Proposition 1.''' For all <math>x,y,z</math> in a group, the following relations hold:
 * <math>(x,yz) = (x,z)(x,y)^z = (x,z)(z,(y,x))(x,y)</math>;
 * <math>(xy,z) = (x,z)^y (y,z) = (x,z)((x,z),y)(y,z)</math>;
@@ Line 45: / Line 47: @@
 The group <math>(A,B)</math> is [[trivial group | trivial]] if and only if <math>A</math> [[centralizer | centralizes]] <math>B</math>.  Also, <math>(A,B) \subseteq A</math> if and only if <math>B</math> [[normalizer | normalizes]] <math>A</math>.  If <math>A</math> and <math>B</math> are both [[normal subgroup | normal]] (or [[characteristic subgroup | characteristic]]), then so is <math>(A,B)</math>, for if <math>f</math> is an ([[inner automorphism |inner]]) [[automorphism]], then
 <cmath> f((a,b)) = (f(a),f(b)). </cmath>
+Note also that since <math>(a,b) = (b,a)^{-1}</math>, <math>(A,B)=(B,A)</math>.
+'''Lemma.''' Let <math>A,B</math> be a closed subsets of <math>G</math> (not necessarily a subgroups); denote by <math>(A,B)</math> the subgroup of <math>G</math> generated by elements of the form <math>(a,b)</math>, for <math>a\in A</math>, <math>b\in B</math>.  Then <math>(A,B)^a \subseteq </math>.
+''Proof.''  Let <math>a,a'</math> be elements of <math>A</math> and <math>b</math> be an element of <math>B</math>.  Then
+<cmath> (a,b)^{a'} = a'^{-1}a^{-1}b^{-1}aba' = (aa',b)b^{-1}a'^{-1}aba' </cmath>
+'''Proposition 2.''' Let <math>A,B,C</math> be three subgroups of <math>G</math>.
+# The group <math>A</math> normalizes the group <math>(A,B)</math>.
+# If the group <math>(B,C)</math> normalizes <math>A</math>, then the set of elements <math>(a,(b,c))</math>, for <math>a\in A</math>, <math>b\in B</math>, <math>c \in C</math>, generates the group <math>(A,(B,C))</math>.
+# If <math>A</math>, <math>B</math>, and <math>C</math> are normal subgroups of <math>G</math>, then <math>(A,(B,C))</math> is a subgroup of the group <math>(C,(B,A)) \cdot (B, (C,A))</math>.
+''Proof.''  For the first, we note that for any <math>a,a' \in A</math> and any <math>b \in B</math>,
+<cmath> (a,b)^{a'}  = (aa',b) (a',b)^{-1}, </cmath>
+by Proposition 1.
+For the second, we have for any <math>x\in G</math>, <math>a\in A</math>, <math>b\in B</math>, <math>c \in C</math>,
+<cmath> \begin{align*}
+(a,(b,c)x) &= a^{-1}x^{-1}(c,b)a(b,c)x = (a,x)x^{-1}a^{-1}(c,b)a(b,c)x \\
+&= (a,x) (x,((b,c),a)) (a,(b,c)) .
+\end{align*} </cmath>
+Since <math>(B,C)</math> normalizes <math>A</math>, the element <math>((b,c),a)</math> lies in <math>A</math>.  It then follows from induction on <math>n</math> that for all <math>b_i \in B</math>, <math>c_i \in C</math>, the element
+<cmath> \biggl( a, \prod_{i=1}^n (b_i,c_i) \biggr) </cmath>
+lies in the subgroup generated by elements of the form <math>(a,(b,c))</math>.  Similarly,
+<cmath> \begin{align*}
+(a,(b,c)) &= (a, (c,b) \cdot (b,c)^2) \\
+&= (a,(b,c)^2)\Bigl( (b,c)^2,\bigl((c,b),a \bigr) \Bigr) \cdot (a,(c,b)) ,
+\end{align*} </cmath>
+lies in the subgroup generated by elements of the form <math>(a,(b,c))</math>; it then follows that <math>(a,(c,b))</math> does.  Then using the observation
+<cmath> (a,(c,b)x) = (a,x)(x, ((c,b),a)) (a,(c,b)), </cmath>
+we prove by induction on <math>n</math> that the element
+<cmath> \biggl(a, \prod_{i=1}^n (b_i,c_i)^{\pm 1} \biggr) </cmath>
+lies in the subgroup generated by elements of the form <math>(a,(b,c))</math>.  This proves the second result.
+For the third part, we first note that since <math>A,B,C</math> are normal subgroups, so are <math>(A,(B,C))</math>, <math>(C,(B,A))</math>, and <math>(B,(C,A))</math>; in particular, <math>(C,(B,A)) \cdot (B,(C,A)) = (B,(C,A)) \cdot (C,(B,A))</math> is a group.  From the previous result of this proposition, it suffices to show that <math>(a,(b,c))</math> lies in this group, for all <math>a\in A</math>, <math>b\in B</math>, and <math>c\in C</math>.  To that end, we note that since <math>A</math> is normal, there exists <math>a' \in A</math> such that <math>a = a'^{b}</math>.  Then by the third result of Proposition 1,
+<cmath> (a'^b , (b,c)) \cdot (b^c , (c,a')) \cdot (c^{a'} , (a',b)) = e, </cmath>
+or
+<cmath> (a,(b,c)) = \bigl[ (b^c, (c,a')) \cdot (c^{a'}} , (a',b)) \bigr]^{-1} \in (B,(C,A)) \cdot (C, (B,A)) , </cmath>
+as desired.  <math>\blacksquare</math>
 == See also ==

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Difference between revisions of "Commutator (group)"

Latest revision as of 23:55, 1 June 2008

Relations with Commutators

Commutators and Subgroups

See also