Groups of order 84: Difference between revisions

Latest revision as of 23:14, 1 August 2011

This article gives information about, and links to more details on, groups of order 84
See pages on algebraic structures of order 84 | See pages on groups of a particular order

Statistics at a glance

The number 84 has the prime factorization:

$\!84=2^{2}\cdot 3^{1}\cdot 7^{1}=4\cdot 3\cdot 7$

All groups of this order are solvable groups.

Quantity	Value	Explanation
Total number of groups up to isomorphism	15
Number of abelian groups (i.e., finite abelian groups) up to isomorphism	2	(number of abelian groups of order $2^{2}$ ) times (number of abelian groups of order $3^{1}$ ) times (number of abelian groups of order $7^{1}$ ) = (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) times (number of unordered integer partitions of 1) = $2\times 1\times 1=2$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism	2	(number of groups of order 4) times (number of groups of order 3) times (number of groups of order 7) = $2\times 1\times 1=2$ .
Number of supersolvable groups (i.e., finite supersolvable groups) up to isomorphism	13
Number of solvable groups (i.e., finite solvable groups) up to isomorphism	15	All groups of this order are solvable.
Number of simple groups up to isomorphism	0	Follows from all groups of this order being solvable.

GAP implementation

The order 84 is part of GAP's SmallGroup library. Hence, any group of order 84 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 84 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(84);

  There are 15 groups of order 84.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 42, 1 ].
     2 has Frattini factor [ 42, 2 ].
     3 has Frattini factor [ 42, 3 ].
     4 has Frattini factor [ 42, 4 ].
     5 has Frattini factor [ 42, 5 ].
     6 has Frattini factor [ 42, 6 ].
     7 - 15 have trivial Frattini subgroup.

  For the selection functions the values of the following attributes
  are precomputed and stored:
     IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
     LGLength, FrattinifactorSize and FrattinifactorId.

  This size belongs to layer 2 of the SmallGroups library.
  IdSmallGroup is available for this size.

@@ Line 7: / Line 7: @@
 <math>\! 84 = 2^2 \cdot 3^1 \cdot 7^1 = 4 \cdot 3 \cdot 7</math>
+All groups of this order are [[solvable group]]s.
+{| class="sortable" border="1"
+! Quantity !! Value !! Explanation
+|-
+| Total number of groups up to isomorphism || [[count::15]] ||
+|-
+| Number of [[abelian group]]s (i.e., [[finite abelian group]]s) up to isomorphism || [[abelian count::2]] || (number of abelian groups of order <math>2^2</math>) times (number of abelian groups of order <math>3^1</math>) times (number of abelian groups of order <math>7^1</math>) = ([[number of unordered integer partitions]] of 2) times ([[number of unordered integer partitions]] of 1) times ([[number of unordered integer partitions]] of 1) = <math>2 \times 1 \times 1 = 2</math>. {{abelian count explanation}}
+|-
+| Number of [[nilpotent group]]s (i.e., [[finite nilpotent group]]s) up to isomorphism || [[nilpotent count::2]] || (number of [[groups of order 4]]) times (number of [[groups of order 3]]) times (number of [[groups of order 7]]) = <math>2 \times 1 \times 1 = 2</math>.
+|-
+| Number of [[supersolvable group]]s (i.e., [[finite supersolvable group]]s) up to isomorphism || [[supersolvable group::13]] ||
+|-
+| Number of [[solvable group]]s (i.e., [[finite solvable group]]s) up to isomorphism || [[solvable count::15]] || All groups of this order are solvable.
+|-
+| Number of simple groups up to isomorphism || 0 || Follows from all groups of this order being solvable.
+|}
 ==GAP implementation==