Direct product of Z4 and Z2

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 defining ingredient::cyclic group:Z4 and defining ingredient::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 $$ma + nb = (m,n)$$.

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:

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


 * 1) The trivial subgroup. Isomorphic to subgroup::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 subgroup::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 subgroup::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 subgroup::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.

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

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

Internal links

 * Subgroup structure of direct product of Z4 and Z2
 * Supergroups of direct product of Z4 and Z2