M16

Definition

The group, sometimes denoted $M_{16}$, is defined as follows:

$M_{16}=⟨a,x\mid a^8=x^2=e,xax=a^5⟩$.

Here, $e$ denotes the identity element.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	2
order (number of elements, equivalently, cardinality or size of underlying set)	16	groups with same order
prime-base logarithm of order	4	groups with same prime-base logarithm of order
max-length of a group	4		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	4		chief length equals prime-base logarithm of order for group of prime power order
composition length	4		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	8	groups with same order and exponent of a group | groups with same exponent of a group	cyclic subgroup of order 8.
prime-base logarithm of exponent	3	groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class	2	groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length	2	groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length	the derived subgroup is contained in the cyclic subgroup and is hence abelian
Frattini length	3	groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same prime-base logarithm of 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 prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group	All proper subgroups are cyclic, dihedral, or Klein four-groups.
rank of a p-group	2	groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group	there exist Klein four-subgroups.
normal rank of a p-group	2	groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group	all abelian normal subgroups are cyclic.
characteristic rank of a p-group	2	groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group	There is a unique (hence characteristic) Klein four-subgroup.

Group properties

Property	Satisfied?	Explanation	Comment
group of prime power order	Yes
nilpotent group	Yes	prime power order implies nilpotent
supersolvable group	Yes	via nilpotent: finite nilpotent implies supersolvable
solvable group	Yes	via nilpotent: nilpotent implies solvable
Abelian group	No	$a,x$ do not commute
Metacyclic group	Yes
Finite group that is 1-isomorphic to an abelian group	Yes	via cocycle halving generalization of Baer correspondence	See element structure of groups of order 16#1-isomorphism

Elements

Further information: element structure of M16

1-isomorphism

The group is 1-isomorphic to the group direct product of Z8 and Z2. In other words, there is a bijection between the groups that restricts to an isomorphism on all cyclic subgroups on either side. The 1-isomorphism is explained by the cocycle halving generalization of Baer correspondence, where the intermediary is a class two Lie cring.

Subgroup structure

Further information: subgroup structure of M16

To describe subgroups, we use the defining presentation given at the beginning:

$M_{16}=⟨a,x\mid a^8=x^2=e,xax=a^5⟩$.

The subgroups are as follows:

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The two-element subgroup $⟨a^4⟩$. This is the derived subgroup, and is also the socle. In particular, it is a characteristic subgroup. Isomorphic to cyclic group:Z2. (1)
3. The two-element subgroups $⟨x⟩$ and $⟨a^{4}x⟩$. These are conjugate subgroups. Isomorphic to cyclic group:Z2. (2)
4. The four-element subgroup $⟨a^2⟩$. This is the center, and is also the Frattini subgroup. In particular, it is a characteristic subgroup. Isomorphic to cyclic group:Z4. (1)
5. The four-element subgroup $⟨a^{2}x⟩$. This is a characteristic subgroup. Isomorphic to cyclic group:Z4. (1)
6. The four-element subgroup $⟨a^4,x⟩$. This is a characteristic subgroup. Isomorphic to Klein four-group. (1)
7. The eight-element subgroups $⟨a⟩$ and $⟨ax⟩$. These are both normal subgroups and are automorphic subgroups -- an outer automorphism interchanges them. Isomorphic to cyclic group:Z8. (2)
8. The eight-element subgroup $⟨a^2,x⟩$. This is a characteristic subgroup. Isomorphic to direct product of Z4 and Z2. (1)
9. The whole group. (1)

GAP implementation

Group ID

This finite group has order 16 and has ID 6 among the groups of order 16 in GAP's SmallGroup library. For context, there are groups of order 16. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(16,6)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(16,6);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [16,6]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.