Why a group is not a direct or semi direct product

In summary, the conversation discusses the question of whether or not \mathbb{Z}_4 can be a direct or semi-direct product of two copies of \mathbb{Z}_2. It is determined that this is not possible because \mathbb{Z}_4 is not isomorphic to \mathbb{Z}_2 and there is a property of an element in \mathbb{Z}_4 that cannot be reproduced by any product of \mathbb{Z}_2.
  • #1
DeldotB
117
8

Homework Statement


Good day all!

(p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out)

So I was wondering if my understanding of this is correct:

The Question asks: "[tex]\mathbb{Z}_4[/tex] has a subgroup is isomorphic to [tex]\mathbb{Z}_2[/tex] The quotient [tex]\mathbb{Z}_4/ \mathbb{Z}_2[/tex] is also isomorphic to [tex]\mathbb{Z}_2[/tex]Nevertheless, [tex]\mathbb{Z}_4[/tex] is not a direct or semidirect product of two copies of [tex]\mathbb{Z}_2[/tex]. Why?

Homework Equations


None

The Attempt at a Solution


Well: Its pretty easy to see that [tex]\mathbb{Z}_{4}\neq \mathbb{Z}_{2}\oplus \mathbb{Z}_{2} [/tex] becuase 2 is not relatively prime to 2. Thus, Z4 isn't a direct product.

For the semi direct product: since [tex]\mathbb{Z}_{2}\cap \mathbb{Z}_{2}\neq e[/tex] (where e is the identity), Z4 is not a semi direct produt of two copies of Z2. Is this correct? (and sufficient?)
 
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  • #2
DeldotB said:
(p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out)
Because this is the definition of that tag. Use [ itex] or double hashes instead if you want to write equations in text.

##\mathbb Z_4## is not isomorphic to ##\mathbb Z_2##. You cannot have a bijection between sets of different cardinality.

Can you think of a property of any element in ##\mathbb Z_4## which none of the products reproduce?
 
  • #3
Orodruin said:
Because this is the definition of that tag. Use [ itex] or double hashes instead if you want to write equations in text.

##\mathbb Z_4## is not isomorphic to ##\mathbb Z_2##. You cannot have a bijection between sets of different cardinality.

Can you think of a property of any element in ##\mathbb Z_4## which none of the products reproduce?
[1] in Z4 has order 4
 
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  • #4
DeldotB said:
<1>
Orodruin said:
Because this is the definition of that tag. Use [ itex] or double hashes instead if you want to write equations in text.

##\mathbb Z_4## is not isomorphic to ##\mathbb Z_2##. You cannot have a bijection between sets of different cardinality.

Can you think of a property of any element in ##\mathbb Z_4## which none of the products reproduce?

I miss typed the question. I meant to say Z4 has a subgroup isomorphic to Z2 (obviously subgroup generated by [2])
 

FAQ: Why a group is not a direct or semi direct product

Why is a group not always a direct or semi direct product?

A group is not always a direct or semi direct product because not all groups can be expressed as a direct or semi direct product of two or more smaller groups. This is because the structure of a group is determined by its group operation, and not all groups have a suitable group operation that allows them to be decomposed into smaller groups.

What is a direct product of groups?

A direct product of groups is a way of combining two or more groups to create a larger group. It is denoted by the symbol "×" and is defined as the cartesian product of the groups with a modified group operation. In a direct product, the elements of the larger group are ordered pairs of elements from the smaller groups, and the group operation is defined component-wise.

How is a semi direct product different from a direct product?

A semi direct product is a generalization of a direct product, where the group operation is not necessarily defined component-wise. In a semi direct product, the elements of the larger group are still ordered pairs of elements from the smaller groups, but the group operation may involve some interplay between the two components. This allows for more flexibility in creating new groups.

Can any group be expressed as a direct or semi direct product?

No, not all groups can be expressed as a direct or semi direct product of smaller groups. For example, simple groups, which cannot be decomposed into non-trivial normal subgroups, cannot be expressed as a direct or semi direct product. Additionally, some groups may have a suitable structure for a direct or semi direct product, but may not have the necessary subgroups to act as the smaller components.

What is an example of a group that is not a direct or semi direct product?

A classic example of a group that is not a direct or semi direct product is the symmetric group on 5 elements (S5). This group has no non-trivial normal subgroups, so it cannot be expressed as a direct or semi direct product. Additionally, there are no two smaller groups that can be combined to create S5, as the only proper subgroups of S5 are A5 and the trivial group.

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