Restricted Burnside group - Revision history

Vipul: /* Definition */

2017-04-11T04:56:08Z

Definition

← Older revision		Revision as of 04:56, 11 April 2017
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	==Definition==		==Definition==

	The '''restricted Burnside group''' <math>RB(d,n)</math>, sometimes denoted <math>R(d.n)</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(d.n)</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(d,n)</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(d,n)</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>(d,n)</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==		==Facts==

Vipul at 06:13, 13 April 2015

2015-04-13T06:13:19Z

← Older revision		Revision as of 06:13, 13 April 2015
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	==Definition==		==Definition==

	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.		The '''restricted Burnside group''' <math>RB(d,n)</math>, sometimes denoted <math>R(d.n)</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(d.n)</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>(d,n)</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''.

Vipul: /* Facts */

2015-04-13T06:12:52Z

Facts

← Older revision		Revision as of 06:12, 13 April 2015
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	* [[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>d</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>d</math>.
	* The condition that this group be finite is ''weaker'' than the condition that the [[Burnside group]] <math>B(d,n)</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.		* The condition that this group be finite is ''weaker'' than the condition that the [[Burnside group]] <math>B(d,n)</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(d,n)</math> is finite, <math>RB(d,n)</math> is isomorphic to <math>B(n,d)</math>.		* In those cases where <math>B(d,n)</math> is finite, <math>RB(d,n)</math> is isomorphic to <math>B(d,n)</math>.

Vipul: /* Facts */

2015-04-13T06:12:42Z

Facts

← Older revision		Revision as of 06:12, 13 April 2015
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	* [[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>d</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>d</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(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.		* The condition that this group be finite is ''weaker'' than the condition that the [[Burnside group]] <math>B(d,n)</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(d,n)</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(d,n)</math> is isomorphic to <math>B(n,d)</math>.

Vipul: /* Facts */

2015-04-13T06:12:15Z

Facts

← Older revision		Revision as of 06:12, 13 April 2015
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	==Facts==		==Facts==

	* [[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>.		* [[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>d</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(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.		* 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(d,n)</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>.

Vipul at 06:12, 13 April 2015

2015-04-13T06:12:07Z

← Older revision		Revision as of 06:12, 13 April 2015
Line 1:		Line 1:
	==Definition==		==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==		==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>.

Vipul at 04:03, 6 June 2012

2012-06-06T04:03:29Z

← Older revision		Revision as of 04:03, 6 June 2012
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	==Definition==		==Definition==

	The '''restricted Burnside group''' <math>RB(n,d)</math>, sometimes denoted <math>R(n,d)</math>, is defined as 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(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.

	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>(n,d)</math> has the answer ''Yes''.

Vipul: /* Facts */

2012-06-06T03:56:00Z

Facts

← Older revision		Revision as of 03:56, 6 June 2012
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	==Facts==		==Facts==

	* 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(n,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(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.
	* 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(n,d)</math> is finite, <math>RB(n,d)</math> is isomorphic to <math>B(n,d)</math>.

Vipul: Created page with "==Definition== The '''restricted Burnside group''' $RB(n,d)$ , sometimes denoted $R(n,d)$ , is defined as the finite residual of the [[Burnside group]..."

2012-06-06T03:55:32Z

Created page with "==Definition== The '''restricted Burnside group''' <math>RB(n,d)</math>, sometimes denoted <math>R(n,d)</math>, is defined as the finite residual of the [[Burnside group]..."

New page

==Definition==

The '''restricted Burnside group''' <math>RB(n,d)</math>, sometimes denoted <math>R(n,d)</math>, is defined as 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.

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''.

==Facts==

* For any [[prime number]] <math>p</math>, the group <math>RB(n,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.
* In those cases where <matH>B(n,d)</math> is finite, <math>RB(n,d)</math> is isomorphic to <math>B(n,d)</math>.

Restricted Burnside group - Revision history

Vipul: /* Definition */

Vipul at 06:13, 13 April 2015

Vipul: /* Facts */

Vipul: /* Facts */

Vipul: /* Facts */

Vipul at 06:12, 13 April 2015

Vipul at 04:03, 6 June 2012

Vipul: /* Facts */

Vipul: Created page with "==Definition== The '''restricted Burnside group''' RB(n,d), sometimes denoted R(n,d), is defined as the finite residual of the [[Burnside group]..."

Vipul: Created page with "==Definition== The '''restricted Burnside group''' $RB(n,d)$ , sometimes denoted $R(n,d)$ , is defined as the finite residual of the [[Burnside group]..."