Direct product of Z4 and Z2

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:

$G : = ⟨ a, b ∣ a^{4} = b^{2} = e, a b = b a ⟩$ .

Writing the presentation in additive notation, with commutativity implicit:

$G : = ⟨ a, b ∣ 4 a = 2 b = 0 ⟩$ .

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 $m a + n b = (m, n)$ .

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

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):

The trivial subgroup. Isomorphic to trivial group. (1)
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)
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)
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)
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)
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.