What am I missing?What is the Proof for Cyclic Groups Being Subgroups?

In summary, The conversation is about understanding a proof given in a link about subgroup of a cyclic group being cyclic. The person is confused about a part where it is concluded that r must be 0. The justification is that m is the smallest integer, forcing r to be 0. The other person clarifies that m is the smallest positive integer such that a^m is in H, and since a^r is also in H, r cannot be any positive integer greater than or equal to 0. The key information is that a^m is in H, so is (a^m)^-1 and all powers of the inverse due to closure.
  • #1
Bashyboy
1,421
5
Hello everyone,

I am trying to understand the proof given in this link:

https://proofwiki.org/wiki/Subgroup_of_Cyclic_Group_is_Cyclic

I understand everything up until the part where they conclude that ##r## must be ##0##. Their justification for this is, that ##m## is the smallest integer, and so this forces ##r=0##. Is it not possible that ##m=3## could be the smallest integer? Couldn't ##r## then be ##2##?
 
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  • #2
No, you are missing the point. m was, by hypothesis, the smallest positive integer such that [itex]a^m\in H[/itex]. We also have that [itex]a^r\in H[/itex]. Once we have that [itex]0\le r< m[/itex], we cannot have r any positive integer because m is the smallest such integer.
 
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  • #3
It seems like the critical information is in this line "Since ## a^m∈H## so is ##(a^m)^{−1}## and all powers of the inverse by closure ."
 

1. What is a cyclic group?

A cyclic group is a mathematical structure that consists of a set of elements and a binary operation (usually denoted as *) that combines two elements to produce a third element. The elements in a cyclic group follow a specific pattern, where each element is a power of a designated generator element. This pattern repeats itself, making the group "cyclic."

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

A group is cyclic if it can be generated by a single element. This means that every element in the group can be written as a power of the generator element. For example, in the group {1, 2, 3, 4, 5}, if we choose 2 as the generator element, then each element in the group can be written as a power of 2: 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8 (which wraps around to 3 in the group), 2^4 = 16 (which wraps around to 1 in the group).

3. What is the order of a cyclic group?

The order of a cyclic group is the number of elements in the group. For a cyclic group generated by a single element, the order is equal to the number of powers that the generator element can have before it repeats itself. In the example from question 2, the order of the group is 5, since there are 5 distinct elements in the group.

4. Can a non-cyclic group have a cyclic subgroup?

Yes, a non-cyclic group can have a cyclic subgroup. A subgroup is a subset of a group that forms a group itself under the same operation. For example, the group of integers under addition is non-cyclic, but it has cyclic subgroups, such as the group {0, 2, 4, 6, 8} under addition modulo 10.

5. What is the significance of cyclic groups in mathematics?

Cyclic groups have many applications in mathematics, particularly in number theory and abstract algebra. They are used to study patterns and symmetries in different mathematical structures, and they have connections to other important concepts such as prime numbers and group theory. In cryptography, cyclic groups are used in algorithms for secure communication and data encryption.

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