Groups of order 168: Difference between revisions

Revision as of 00:13, 31 July 2011

This article gives information about, and links to more details on, groups of order 168
See pages on algebraic structures of order 168 | See pages on groups of a particular order

Statistics at a glance

The prime factorization of 168 is:

$168 = 2^{3} \cdot 3 \cdot 7 = 8 \cdot 3 \cdot 7$

Quantity	Value	List/comment
Total number of groups	57
Total number of abelian groups	3	(number of abelian groups of order $2^{3}$ ) times (number of abelian groups of order $3^{1}$ ) times (number of abelian groups of order $7^{1}$ ) = $3 \times 1 \times 1 = 3$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Total number of nilpotent groups	5	(number of groups of order 8) times (number of groups of order 3) times (number of groups of order 7) = $5 \times 1 \times 1 = 5$ . See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.
Total number of solvable groups	56	the only non-solvable group is the simple non-abelian group projective special linear group:PSL(3,2), which is also isomorphic to $P S L (2, 7)$ .
Total number of simple groups	1	the simple non-abelian group projective special linear group:PSL(3,2), which is also isomorphic to $P S L (2, 7)$ .

@@ Line 9: / Line 9: @@
 ! Quantity !! Value !! List/comment
 |-
-| Total number of groups || 57 ||
+| Total number of groups || [[count::57]] ||
 |-
-| Total number of [[abelian group]]s || 3 || ((number of abelian groups of order 8) = 3) times (number of abelian groups of order 3) = 1) times (number of abelian groups of order 7) = 1). See [[classification of finite abelian groups]] and [[structure theorem for finitely generated abelian groups]].
+| Total number of [[abelian group]]s || [[abelian count::3]] || (number of abelian groups of order <math>2^3</math>) times (number of abelian groups of order <math>3^1</math>) times (number of abelian groups of order <math>7^1</math>) = <math>3 \times 1 \times 1 = 3</math>. {{abelian count explanation}}
 |-
-| Total number of [[nilpotent group]]s || 5 || ((number of [[groups of order 8]]) = 5) times ((number of [[groups of order 3]]) = 1) times ((number of [[groups of order 5]]) = 1). See [[equivalence of definitions of finite nilpotent group]]
+| Total number of [[nilpotent group]]s || [[nilpotent count::5]] || (number of [[groups of order 8]]) times (number of [[groups of order 3]]) times (number of [[groups of order 7]]) = <math>5 \times 1 \times 1 = 5</math>. {{nilpotent count explanation}}
 |-
-| Total number of [[solvable group]]s || 56 || the only ''non-solvable'' group is the [[simple non-abelian group]] [[projective special linear group:PSL(3,2)]], which is also isomorphic to <math>PSL(2,7)</math>.
+| Total number of [[solvable group]]s || [[solvable count::56]] || the only ''non-solvable'' group is the [[simple non-abelian group]] [[projective special linear group:PSL(3,2)]], which is also isomorphic to <math>PSL(2,7)</math>.
 |-
 | Total number of [[simple group]]s || 1 || the [[simple non-abelian group]] [[projective special linear group:PSL(3,2)]], which is also isomorphic to <math>PSL(2,7)</math>.
 |}