Restricted Burnside group: Difference between revisions

Revision as of 06:12, 13 April 2015

Definition

The restricted Burnside group $R B (d, n)$ , sometimes denoted $R (n, d)$ , is defined as the quotient group by the finite residual of the Burnside group $B (d, n)$ . In other words, it is the quotient of $B (n, d)$ by the intersection of all normal subgroups of finite index in it.

Note that this group is a finite group if and only if the restricted Burnside problem for the pair $(d, n)$ has the answer Yes.

Facts

Kostrikin's theorem on restricted Burnside problem: For any prime number $p$ , the group $R B (d, p)$ is finite for every value of $n$ .
The condition that this group be finite is weaker than the condition that the Burnside group $B (n, d)$ be finite, i.e., there are many cases where $B (d, n)$ is known to be infinite and $R B (d, n)$ is known to be finite. This includes all odd primes greater than 665.
In those cases where $B (d, n)$ is finite, $R B (n, d)$ is isomorphic to $B (n, d)$ .

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 ==Definition==
-The '''restricted Burnside group''' <math>RB(n,d)</math>, sometimes denoted <math>R(n,d)</math>, is defined as the [[quotient group]] by the [[finite residual]] of the [[Burnside group]] <math>B(n,d)</math>. In other words, it is the quotient of <math>B(n,d)</math> by the intersection of all [[normal subgroup of finite index|normal subgroups of finite index]] in it.
+The '''restricted Burnside group''' <math>RB(d,n)</math>, sometimes denoted <math>R(n,d)</math>, is defined as the [[quotient group]] by the [[finite residual]] of the [[Burnside group]] <math>B(d,n)</math>. In other words, it is the quotient of <math>B(n,d)</math> by the intersection of all [[normal subgroup of finite index|normal subgroups of finite index]] in it.
-Note that this group is a [[finite group]] if and only if the [[restricted Burnside problem]] for the pair <math>(n,d)</math> has the answer ''Yes''.
+Note that this group is a [[finite group]] if and only if the [[restricted Burnside problem]] for the pair <math>(d,n)</math> has the answer ''Yes''.
 ==Facts==
-* [[Kostrikin's theorem on restricted Burnside problem]]: For any [[prime number]] <math>p</math>, the group <math>RB(n,p)</math> is finite for every value of <math>n</math>.
+* [[Kostrikin's theorem on restricted Burnside problem]]: For any [[prime number]] <math>p</math>, the group <math>RB(d,p)</math> is finite for every value of <math>n</math>.
-* The condition that this group be finite is ''weaker'' than the condition that the [[Burnside group]] <math>B(n,d)</math> be finite, i.e., there are many cases where <math>B(n,d)</math> is known to be infinite and <math>RB(n,d)</matH> is known to be finite. This includes all odd primes greater than 665.
+* The condition that this group be finite is ''weaker'' than the condition that the [[Burnside group]] <math>B(n,d)</math> be finite, i.e., there are many cases where <math>B(d,n)</math> is known to be infinite and <math>RB(d,n)</math> is known to be finite. This includes all odd primes greater than 665.
-* In those cases where <matH>B(n,d)</math> is finite, <math>RB(n,d)</math> is isomorphic to <math>B(n,d)</math>.
+* In those cases where <math>B(d,n)</math> is finite, <math>RB(n,d)</math> is isomorphic to <math>B(n,d)</math>.