Difference between revisions of "Dicyclic group:Dic20"
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Alternatively, it has the presentation: | Alternatively, it has the presentation: | ||
− | <math>\langle a,b,c \mid a^{10} = b^ | + | <math>\langle a,b,c \mid a^{10} = e, a^5 = b^2, bab^{-1} = a^{-1} \rangle</math>. |
==Arithmetic functions== | ==Arithmetic functions== | ||
− | {| class=" | + | {| class="sortable" border="1" |
− | ! Function !! Value !! Explanation | + | ! Function !! Value !! Similar groups !! Explanation |
|- | |- | ||
− | | | + | | {{arithmetic function value order|20}} || As dicyclic group of degree <math>m = 5</math>: <math>4m = 4(5) = 20</math> |
|- | |- | ||
− | | | + | | {{arithmetic function value given order|exponent of a group|10|20}} || As dicyclic group of degree <math>m = 5</math>: <math>2m = 2(5) = 10</math>. |
|- | |- | ||
− | | [[Frattini | + | | {{arithmetic function value given order|Frattini length|2|20}} || The [[Frattini subgroup]] is isomorphic to [[cyclic group:Z2]] -- specifically, it is the center. |
|- | |- | ||
− | | | + | | {{arithmetic function value given order|derived length|2|20}} || The group is in fact a [[metacyclic group]], hence metabelian, but it is not abelian. |
|- | |- | ||
− | | [[nilpotency class]] || -- || Not a nilpotent group. | + | | [[nilpotency class]] || -- || -- || Not a nilpotent group. |
|- | |- | ||
− | | | + | | {{arithmetic function value given order|minimum size of generating set|2|20}} || |
|- | |- | ||
− | | | + | | {{arithmetic function value given order|subgroup rank of a group|2|20}} || |
|} | |} | ||
==Group properties== | ==Group properties== | ||
− | {| class=" | + | ===Important properties=== |
− | ! Property !! Satisfied !! Explanation | + | |
+ | {| class="sortable" border="1" | ||
+ | ! Property !! Satisfied? !! Explanation | ||
|- | |- | ||
− | | [[cyclic group]] || No || | + | | [[dissatisfies property::cyclic group]] || No || |
|- | |- | ||
− | | [[abelian group]] || No || | + | | [[dissatisfies property::abelian group]] || No || |
|- | |- | ||
− | | [[nilpotent group]] || No || | + | | [[dissatisfies property::nilpotent group]] || No || |
|- | |- | ||
− | | [[metacyclic group]] || Yes || | + | | [[satisfies property::metacyclic group]] || Yes || |
+ | |- | ||
+ | | [[satisfies property::supersolvable group]] || Yes || | ||
+ | |- | ||
+ | | [[satisfies property::solvable group]] || Yes || | ||
+ | |} | ||
+ | |||
+ | ===Other properties=== | ||
+ | {| class="sortable" border="1" | ||
+ | ! Property !! Satisfied? !! Explanation | ||
|- | |- | ||
− | | [[ | + | | [[dissatisfies property::ambivalent group]] || No || |
|- | |- | ||
− | | [[ | + | | [[satisfies property::Schur-trivial group]] || Yes || This is true for all [[dicyclic group]]s. |
|- | |- | ||
− | | [[ | + | | [[satisfies property::finite group with periodic cohomology]] || Yes || This is true for all [[dicyclic group]]s. |
|} | |} | ||
Latest revision as of 01:53, 16 January 2013
This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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Contents
Definition
This group is defined as the dicyclic group of order , and hence degree
. In other words, it has the presentation:
Alternatively, it has the presentation:
.
Arithmetic functions
Function | Value | Similar groups | Explanation |
---|---|---|---|
order (number of elements, equivalently, cardinality or size of underlying set) | 20 | groups with same order | As dicyclic group of degree ![]() ![]() |
exponent of a group | 10 | groups with same order and exponent of a group | groups with same exponent of a group | As dicyclic group of degree ![]() ![]() |
Frattini length | 2 | groups with same order and Frattini length | groups with same Frattini length | The Frattini subgroup is isomorphic to cyclic group:Z2 -- specifically, it is the center. |
derived length | 2 | groups with same order and derived length | groups with same derived length | The group is in fact a metacyclic group, hence metabelian, but it is not abelian. |
nilpotency class | -- | -- | Not a nilpotent group. |
minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set | |
subgroup rank of a group | 2 | groups with same order and subgroup rank of a group | groups with same subgroup rank of a group |
Group properties
Important properties
Property | Satisfied? | Explanation |
---|---|---|
cyclic group | No | |
abelian group | No | |
nilpotent group | No | |
metacyclic group | Yes | |
supersolvable group | Yes | |
solvable group | Yes |
Other properties
Property | Satisfied? | Explanation |
---|---|---|
ambivalent group | No | |
Schur-trivial group | Yes | This is true for all dicyclic groups. |
finite group with periodic cohomology | Yes | This is true for all dicyclic groups. |
GAP implementation
Group ID
This finite group has order 20 and has ID 1 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(20,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(20,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [20,1]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.