Cyclic group has 3 subgroups, what is the order of G

In summary, the conversation discusses the order of a cyclic group with only three distinct subgroups: the identity element, the group itself, and a subgroup of order 5. Using Lagrange's theorem, it is determined that the order of the group must be a multiple of 5. The conversation then explores the possibility of the order being 10, but it is concluded that this cannot be correct as it would require a subgroup of order 2, which is not possible for a cyclic group. Ultimately, it is determined that the order of the group must be 25. The conversation also briefly mentions the fact that for cyclic groups, Lagrange's theorem works both ways.
  • #1
Mr Davis 97
1,462
44

Homework Statement


Suppose a cyclic group, G, has only three distinct subgroups: e, G itself,
and a subgroup of order 5. What is |G|? What if you replace 5 by p where
p is prime?

Homework Equations

The Attempt at a Solution


So, G has three distinct subgroups. By Lagrange's theorem, the order of the subgroup has to divide the order of the group. So the order of G is a multiple of 5. If we let |G| = 5, then there are only two subgroups, G and e. So we try |G| = 10. Why couldn't 10 be correct? Am I neglecting the fact that G is cyclic? (I know that the answer is actually 25, but am not sure why).
 
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  • #2
Mr Davis 97 said:
Why couldn't 10 be correct?

The wording of the problem implies that G can't have a subgroup of order 2.
 
  • #3
Stephen Tashi said:
The wording of the problem implies that G can't have a subgroup of order 2.
But if G has order 10 does that mean that it must necessarily have a subgroup of order 2 and a subgroup of order 5? I thought that lagranges theorem was just an if statement and not an if and only if statement.
 
  • #4
For cyclic subgroups, Lagranges Theorem does work both ways. If k divides |G|, then {e, a^k, a^(2k), ... a^(|G| -k }, is a subgroup of G
 
  • #5
A cyclic group ##G## is Abelian, so all subgroups ##U## are normal subgroups and thus a direct factor. This means ##G/U## is also (isomorphic to) a subgroup ##V## of ##G##.
 
  • #6
Mr Davis 97 said:
But if G has order 10 does that mean that it must necessarily have a subgroup of order 2 and a subgroup of order 5? I thought that lagranges theorem was just an if statement and not an if and only if statement.

A finite group of even order must have an element of order 2: Partition the group into subsets [itex]\{a, a^{-1}\}[/itex]. Such a set contains two elements if [itex]a^2 \neq e[/itex] or one element if [itex]a^2 = e[/itex].

One of these subsets is [itex]\{e\}[/itex] but the group order is even, so there must be at least one other subset consisting of a single element.
 

1. What is a cyclic group?

A cyclic group is a mathematical structure that consists of a set of elements and an operation that combines two elements to produce a third element. The elements in a cyclic group are generated by repeatedly applying the operation to a single element, known as the generator.

2. How do you determine if a group is cyclic?

A group is cyclic if there exists an element in the group that can be used to generate all other elements by repeatedly applying the group operation. In other words, every element in a cyclic group can be written as a power of the generator.

3. What is the order of a group?

The order of a group is the number of elements in the group. In a cyclic group, the order is equal to the number of elements generated by the generator. This can be found by taking the smallest positive integer k such that g^k = e, where g is the generator and e is the identity element.

4. How do you find the order of a subgroup?

The order of a subgroup is equal to the number of elements in the subgroup. To find the order of a subgroup, you can use Lagrange's theorem which states that the order of a subgroup must divide the order of the larger group. Additionally, in a cyclic group, the order of a subgroup is equal to the number of elements generated by the subgroup's generator.

5. How many subgroups can a cyclic group of order 3 have?

A cyclic group of order 3 can have 3 subgroups: the trivial subgroup consisting of only the identity element, a subgroup generated by an element of order 3, and a subgroup generated by an element of order 1. These are the only possible subgroups in a cyclic group of order 3, as every subgroup must have an order that divides the order of the larger group (in this case, 3).

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