# Direct product of Z4 and Z2

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## Definition

### As a direct product

The direct product of Z4 and Z2 is an abelian group of order eight obtained as the external direct product of cyclic group:Z4 and cyclic group:Z2.

### As a presentation

If we denote by $a$ and $b$ the generators of the direct factors, then the presentation is given by: $G := \langle a,b \mid a^4 = b^2 = e, ab = ba \rangle$.

Writing the presentation in additive notation, with commutativity implicit: $G := \langle a,b \mid 4a = 2b = 0 \rangle$.

### Multiplication table

Here, we use ordered pairs, as is typical for external direct products, with the first coordinate corresponding to the cyclic group of order four and the second coordinate corresponding to the cyclic group of order two. From the presentation notation, $a = (1,0), b = (0,1)$ and $ma + nb = (m,n)$.

Element $(0,0)$ $(1,0)$ $(2,0)$ $(3,0)$ $(0,1)$ $(1,1)$ $(2,1)$ $(3,1)$ $(0,0)$ $(0,0)$ $(1,0)$ $(2,0)$ $(3,0)$ $(0,1)$ $(1,1)$ $(2,1)$ $(3,1)$ $(1,0)$ $(1,0)$ $(2,0)$ $(3,0)$ $(0,0)$ $(1,1)$ $(2,1)$ $(3,1)$ $(0,1)$ $(2,0)$ $(2,0)$ $(3,0)$ $(0,0)$ $(1,0)$ $(2,1)$ $(3,1)$ $(0,1)$ $(1,1)$ $(3,0)$ $(3,0)$ $(0,0)$ $(1,0)$ $(2,0)$ $(3,1)$ $(0,1)$ $(1,1)$ $(2,1)$ $(0,1)$ $(0,1)$ $(1,1)$ $(2,1)$ $(3,1)$ $(0,0)$ $(1,0)$ $(2,0)$ $(3,0)$ $(1,1)$ $(1,1)$ $(2,1)$ $(3,1)$ $(0,1)$ $(1,0)$ $(2,0)$ $(3,0)$ $(0,0)$ $(2,1)$ $(2,1)$ $(3,1)$ $(0,1)$ $(1,1)$ $(2,0)$ $(3,0)$ $(0,0)$ $(1,0)$ $(3,1)$ $(3,1)$ $(0,1)$ $(1,1)$ $(2,1)$ $(3,0)$ $(0,0)$ $(1,0)$ $(2,0)$

## As an abelian group of prime power order

The group is a 2-group corresponding to the partition: $\! 3 = 2 + 1$

In other words, it is the group $\mathbb{Z}_{p^2} \times \mathbb{Z}_p$ for the case $p = 2$. Other particular cases include:

Value of prime number $p$ Corresponding group
Generic prime direct product of cyclic group of prime-square order and cyclic group of prime order
3 direct product of Z9 and Z3
5 direct product of Z25 and Z5
7 direct product of Z49 and Z7

## Arithmetic functions

Function Value Similar groups Explanation
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 8 groups with same order
prime-base logarithm of order 3 groups with same prime-base logarithm of order
exponent of a group 4 groups with same order and exponent of a group | groups with same prime-base logarithm of order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 2 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
Frattini length 2 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
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 same as minimum size of generating set since it is an abelian group of prime power order
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 same as minimum size of generating set since it is an abelian group of prime power order
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 same as minimum size of generating set since it is an abelian group of prime power order
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 same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class 1 The group is a nontrivial abelian group
derived length 1 The group is a nontrivial abelian group
Fitting length 1 The group is a nontrivial abelian group

## Group properties

Property Satisfied Explanation Comment
Group of prime power order Yes By definition
Abelian group Yes Direct product of abelian groups
Cyclic group No No element of order eight
Elementary abelian group No Element of order four Smallest abelian group that's not cyclic or elementary abelian
Nilpotent group Yes Abelian implies nilpotent
T-group Yes Abelian groups are T-groups

## Subgroups

Further information: Subgroup structure of direct product of Z4 and Z2

The group has the following eight subgroups (all of which are normal subgroups, since the group is abelian):

1. The trivial subgroup. Isomorphic to trivial group. (1)
2. The cyclic subgroup of order two comprising the squares, i.e., the first agemo subgroup. In our notation, this is the subgroup $\{ (2,0), (0,0 \}$. Isomorphic to cyclic group:Z2. (1)
3. Two other cyclic subgroups of order two, generated by elements that are not squares. In our notation, these are $\{ (0,1), (0,0) \}$ and $\{ (2,1), (0,0) \}$. These are related by an outer automorphism. Isomorphic to cyclic group:Z2. (2)
4. The group of order four comprising all the elements of order dividing two. In other words, the first omega subgroup. In our notation, this is $\{ (2,0), (0,1), (2,1), (0,0) \}$. Isomorphic to Klein four-group. (1)
5. The two cyclic subgroups of order four, generated by elements of order four. In our notation. these are $\{ (1,0), (2,0), (3,0), (0,0) \}$ and $(1,1), (2,0), (3,1), (0,0) \}$. These are related by an outer automorphism. Isomorphic to cyclic group:Z4. (2)
6. The whole group. (1)

### Normal subgroups

Since the group is abelian, all subgroups are normal.

### Characteristic subgroups

The subgroups of type (1), (2), (4) and (6) are characteristic. In particular, there is exactly one characteristic subgroup of each order.

## GAP implementation

### Group ID

This finite group has order 8 and has ID 2 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(8,2)

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

gap> G := SmallGroup(8,2);

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

IdGroup(G) = [8,2]

or just do:

IdGroup(G)

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

### Other descriptions

It can also be described as a direct product using GAP's DirectProduct function:

DirectProduct(CyclicGroup(4),CyclicGroup(2))