Groups of order 60 and elements of order 5

In summary, the conversation discusses the problem of showing that H, the set of all elements in a group G that can be written as a product of elements of order 5, is a normal subgroup of G. After proving that H is indeed a subgroup, the conversation goes on to use the fact that conjugation preserves order and the number of elements in all the Sylow-5 subgroups to show that H must be normal.
  • #1
djxl
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Homework Statement



Let G be a group with order [tex]\left| G \right| = 60[/tex]. Assume that G is simple.

Now let H be the set of all elements that can be written as a product of elements of order 5 in G. Show that H is a normal subgroup of G. Then conclude that H = G

Homework Equations




The Attempt at a Solution



I started by proving that H acutally is a subgroup.

I've then shown that there are 6 Sylow-5 subgroups in G and that they are cyclic. I know that all the elements of order 5 are the generators of the Sylow-5 subgroups. But how I can use that to show that H is normal escapes me.

All help/hints appreciated :).
 
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  • #2
Sylow, won't help you, I don't think - the elements of order 5 do not generate Sylow-5 subgroups. The product of two elements of order 5 is not necessarily an element of order 5 (or any power of 5).

H is trivially a subgroup - there is nothing to prove there. What about normality? This is straight forward - conjugation preserves order, and notice that

gxyg^{-1} = gxg^{1-}gyg^{-1}
 
  • #3
Count the total number of elements in all the 5-sylow groups. Use that number to show that the subgroup must be normal
 
  • #4
Thanks for the quick help. I understand the solution now o:).
 

1. What is the significance of groups of order 60 and elements of order 5 in mathematics?

Groups of order 60 and elements of order 5 are important concepts in abstract algebra, particularly in the study of finite groups. They are used to understand the structure and properties of groups, and have applications in various fields such as cryptography and physics.

2. How many groups of order 60 and elements of order 5 exist?

There are multiple groups of order 60 and elements of order 5, as the specific structure and elements of a group depend on its composition and operations. However, all groups of order 60 and elements of order 5 share certain properties and can be classified into isomorphism classes.

3. What are some examples of groups of order 60 and elements of order 5?

Some examples of groups of order 60 and elements of order 5 include the symmetric group S5, the alternating group A5, and the dihedral group D12. These groups have different compositions and operations, but all have order 60 and elements of order 5.

4. How do groups of order 60 and elements of order 5 relate to other mathematical concepts?

Groups of order 60 and elements of order 5 are closely related to other concepts in mathematics, such as factorization of integers, permutations, and symmetry. They also have connections to other branches of mathematics, including number theory, geometry, and topology.

5. Are there any real-world applications of groups of order 60 and elements of order 5?

Yes, there are real-world applications of groups of order 60 and elements of order 5 in fields such as cryptography, coding theory, and particle physics. For example, the properties of these groups are used in coding and decoding messages securely, and in understanding the symmetries of subatomic particles.

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