Alternating group:A8: Difference between revisions

Revision as of 15:49, 17 September 2011

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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined in the following equivalent ways:

It is the alternating group of degree eight, i.e., over a set of size eight.
It is the projective special linear group of degree four over the field of two elements, i.e., $P S L (4, 2)$ . It is also the special linear group $S L (4, 2)$ , the projective general linear group $P G L (4, 2)$ , and the general linear group $G L (4, 2)$ .

This is one member of the smallest order pair of non-isomorphic finite simple non-abelian groups having the same order. The other member of this pair is projective special linear group:PSL(3,4).

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	20160	groups with same order	As alternating group: $8! / 2 = (8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 2 \cdot 3 \cdot 1) / 2 = 20160$ As general linear group: $(2^{4} - 1) (2^{4} - 2) (2^{4} - 2^{2}) (2^{4} - 2^{3}) = 15 \cdot 14 \cdot 12 \cdot 8 = 20160$
exponent of a group	420	groups with same order and exponent of a group \| groups with same exponent of a group
derived length	--	--	not a solvable group
nilpotency class	--	--	not a nilpotent group
Frattini length	1	groups with same order and Frattini length \| groups with same Frattini length	Frattini-free group: intersection of all maximal subgroups is trivial
minimum size of generating set	2	groups with same order and minimum size of generating set \| groups with same minimum size of generating set

GAP implementation

Description	Functions used
`AlternatingGroup(8)`	AlternatingGroup
`PSL(4,2)`	PSL
`SL(4,2)`	SL
`PGL(4,2)`	PGL
`GL(4,2)`	GL

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 | {{arithmetic function value order|20160}} || As alternating group: <math>8!/2 = (8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 2 \cdot 3 \cdot 1)/2 = 20160</math><br>As general linear group: <math>(2^4 - 1)(2^4 - 2)(2^4 - 2^2)(2^4 - 2^3) = 15 \cdot 14 \cdot 12 \cdot 8 = 20160</math>
+|-
+| {{arithmetic function value given order|exponent of a group|420|20160}} ||
+|-
+| [[derived length]] || -- || -- || not a [[solvable group]]
+|-
+| [[nilpotency class]] || -- || -- || not a [[nilpotent group]]
+|-
+| {{arithmetic function value given order|Frattini length|1|20160}} || [[Frattini-free group]]: intersection of all maximal subgroups is trivial
+|-
+| {{arithmetic function value given order|minimum size of generating set|2|20160}} ||
 |}