Group of order 100 with no element of order 4?

In summary, a group with order 100 and no element of order 4 can be constructed using direct products, such as \mathbb{Z}_{5}\times\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}. This is possible due to the fact that for every prime divisor p of a group, there exists an element with order p.
  • #1
kpoltorak
15
0
1. Homework Statement [/]
Is there a group G with order 100 such that it has no element of order 4? How would one go about proving the existence of such a group?



2. Homework Equations [/]
For every prime divisor p of a group, there exists an element with order p.



The Attempt at a Solution

 
Physics news on Phys.org
  • #2
Do you know about direct products? If so, you should be able to construct such a group pretty easily.
 
  • #3
Yes I do have some knowledge of direct products. Could I construct a group as such: [tex]\mathbb{Z}_{5}\times\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}[/tex]?
 
  • #4
kpoltorak said:
Yes I do have some knowledge of direct products. Could I construct a group as such: [tex]\mathbb{Z}_{5}\times\mathbb{Z}_{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}[/tex]?

That's the one I had in mind.
 

FAQ: Group of order 100 with no element of order 4?

What is a "group of order 100 with no element of order 4"?

A group of order 100 is a mathematical structure consisting of 100 elements that follow specific rules of operation. The phrase "no element of order 4" means that there is no element within the group that, when multiplied by itself 4 times, equals the identity element. In other words, there is no element in the group that has a cyclic subgroup of order 4.

Can a group of order 100 have elements of order 4?

No, by definition, a group of order 100 with no element of order 4 cannot have elements of order 4. This is because if there were an element of order 4, it would form a cyclic subgroup of order 4, contradicting the original condition of the group.

3. What is the significance of having no element of order 4 in a group of order 100?

The lack of an element of order 4 in a group of order 100 has significant implications in the structure and properties of the group. It means that the group does not have any cyclic subgroups of order 4, which can affect the group's symmetry and other characteristics.

4. How many elements of order 2 can a group of order 100 with no element of order 4 have?

A group of order 100 with no element of order 4 can have at most 3 elements of order 2. This is because if there were 4 or more elements of order 2, they would form a cyclic subgroup of order 4, which contradicts the original condition of the group.

5. Is a group of order 100 with no element of order 4 a simple group?

No, a group of order 100 with no element of order 4 is not a simple group. A simple group is a group that has no non-trivial normal subgroups, but in this case, the group has a non-trivial normal subgroup of order 2.

Similar threads

Group Theory: Finite Abelian Groups - An element of order
Replies
3
Views
2K
Understanding the Dicyclic Group of Order 12: Composition and Element Orders
Replies
3
Views
1K
Does Group Cardinality Determine Element Order?
Replies
2
Views
1K
Show that ##G\simeq \mathbb{Z}/2p\mathbb{Z}##
Replies
2
Views
1K
Proving that an Abelian group of order pq is isomorphic to Z_pq
Replies
5
Views
2K
Probability of girls on a team getting shirts with pair numbers
Replies
3
Views
1K
Proof of Subgroup Property for Cyclic Group G: Homework Help
Replies
3
Views
1K
Normal group of order 60 isomorphic to A_5
Replies
6
Views
1K
Isomorphism to certain Galois group and cyclic groups
Replies
1
Views
1K
Showing that a subgroup of Sym(4) is isomorphic to D_8
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top