Question on subgroup and order of the elements

In summary: Since H is a subgroup of G, it means that we must have |H|=1, |H|=2, |H|=7, or |H|=14, mustn't we?Yes.
  • #1
himynameJEF
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Let G be the group of symmetries (including flips) of the regular heptagon (7-gon).

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As usual, we regard the elements of G as permutations of the set of vertex labels; thus, G ≤ S7.

(a) Let σ denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic subgroup R := ⟨σ⟩ as a set of elements of S7 in cycle notation.

(b) What are the orders of each of the elements of R?

does this mean R := ⟨( 1 2 3 4 5 6 7 )⟩?
and I am unsure how to answer part b)

thanks
 

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  • #2
himynameJEF said:
Let G be the group of symmetries (including flips) of the regular heptagon (7-gon).
As usual, we regard the elements of G as permutations of the set of vertex labels; thus, G ≤ S7.

(a) Let σ denote the rotation of the 7-gon that takes the vertex 1 to the vertex 2. Write down the cyclic subgroup R := ⟨σ⟩ as a set of elements of S7 in cycle notation.

(b) What are the orders of each of the elements of R?

does this mean R := ⟨( 1 2 3 4 5 6 7 )⟩?
and I am unsure how to answer part b)

thanks

Hi himynameJEF, welcome to MHB! ;)

Yes, that is correct for what $R$ stands for.
Which elements exactly are in $R$ though?
That is, what is for instance ( 1 2 3 4 5 6 7 )2?

As for (b), let's start with the order of ( 1 2 3 4 5 6 7 ).
What is it?
 
  • #3
I like Serena said:
Hi himynameJEF, welcome to MHB! ;)

Yes, that is correct for what $R$ stands for.
Which elements exactly are in $R$ though?
That is, what is for instance ( 1 2 3 4 5 6 7 )2?

As for (b), let's start with the order of ( 1 2 3 4 5 6 7 ).
What is it?

hi! :)

( 1 2 3 4 5 6 7 )2 would be ( 1 3 5 7 2 4 6 )?

and that would be an order of 7?

thanks!
 
  • #4
himynameJEF said:
hi! :)

( 1 2 3 4 5 6 7 )2 would be ( 1 3 5 7 2 4 6 )?

and that would be an order of 7?

thanks!

Yep. (Nod)

So now we have 2 elements in $R$ that both have order 7.
How about ( 1 2 3 4 5 6 7 )3? (Wondering)
 
  • #5
I like Serena said:
Yep. (Nod)

So now we have 2 elements in $R$ that both have order 7.
How about ( 1 2 3 4 5 6 7 )3? (Wondering)

thanks i understand it now! :)

also another question

Let H be any subgroup of G other than G itself. explain why H is cyclic?

since G is prime then |H| is 1 or 7. then H must equal G and it would be cyclic but the question says any other subgroup other than G so H must be {e}? is this cyclic? I am confused
 
  • #6
himynameJEF said:
thanks i understand it now! :)

also another question

Let H be any subgroup of G other than G itself. explain why H is cyclic?

since G is prime then |H| is 1 or 7. then H must equal G and it would be cyclic but the question says any other subgroup other than G so H must be {e}? is this cyclic? I am confused

Isn't the order of G 14?
That is, aren't there 14 symmetries with rotations and flips?
 
  • #7
I like Serena said:
Isn't the order of G 14?
That is, aren't there 14 symmetries with rotations and flips?

oh! yes sorry your right
so how would i explain this? and then determine the number of subgroups?

thanks!
 
  • #8
himynameJEF said:
oh! yes sorry your right
so how would i explain this? and then determine the number of subgroups?

thanks!

Since H is a subgroup of G, it means that we must have |H|=1, |H|=2, |H|=7, or |H|=14, mustn't we?
Which are the subgroups that correspond to those?
 

Related to Question on subgroup and order of the elements

1. What is a subgroup?

A subgroup is a subset of a larger group that still follows the same group operation and contains an identity element and inverse elements.

2. How is the order of an element in a subgroup determined?

The order of an element in a subgroup is determined by the number of times the element must be combined with itself to get the identity element of the subgroup.

3. Can the order of an element in a subgroup be greater than the order of the subgroup itself?

Yes, the order of an element in a subgroup can be greater than the order of the subgroup itself. This is because the subgroup only needs to contain elements that follow the same group operation, but the element can have a higher order if it is combined with other elements from the larger group.

4. How can we find the subgroups of a given group?

To find the subgroups of a given group, we can use the subgroup test, which involves checking if the subset follows the same group operation and contains the identity element and inverse elements. We can also use the Lagrange's theorem to find all possible subgroups of a finite group.

5. Can a subgroup have a different structure from the larger group?

Yes, a subgroup can have a different structure from the larger group. This means that the subgroup can have a different set of elements and a different group operation, as long as it still follows the requirements of a subgroup (containing an identity element and inverse elements).

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