Determining if Subset is a Subgroup by using Group Presentation

In summary, we discussed two questions about subgroups in a group presentation. For a subset S to be a subgroup of G, the generators of S must satisfy the condition that for any a and b in S, ab^-1 is also in S. Additionally, for A to be a subgroup of G, the generators of A must be able to be written as words in B, the generators of B.
  • #1
Bacle
662
1
Hi, Algebraists:

Say I'm given a group's presentation G=<X|R>, with

X a finite set of generators, R the set of relations. A couple of questions, please:

i)If S is a subset of G what condition must the generators of

S satisfy for S to be a subgroup of G ? I know there is a condition

that if for any a,b in S, then S is a subgroup of G if ab^-1 is in S, but

I am tryng to work only with the generating set.

ii) If A,B are known to be subgroups of G; G as above: what

condition do I need on the generators of A,B respectively,

in order to tell if A is a subgroup of G? Is inclusion enough?

Thanks.
 
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  • #2
I think a subset of a group which is generated by a set of generators is automatically a subgroup. I may be wrong though.
 
  • #3
I got this one: a necessary and sufficient condition is that every generator of A can be
written as a word in B.
 

What is the definition of a subgroup?

A subgroup is a subset of a group that satisfies the same group axioms as the larger group. In other words, it is a smaller group that is contained within a larger group.

What is a group presentation?

A group presentation is a way of describing a group using generators and relations. Generators are elements that can be used to create all the other elements in the group, and relations are equations that must be satisfied by the elements in the group.

How do you determine if a subset is a subgroup?

To determine if a subset is a subgroup, you must show that it satisfies the same group axioms as the larger group. This can be done by checking if the subset contains the identity element, is closed under the group operation, and has inverses for all its elements.

Can a subset be a subgroup of multiple groups?

Yes, a subset can be a subgroup of multiple groups as long as it satisfies the group axioms for each of those groups. However, the group presentation may be different for each group.

What is the significance of determining if a subset is a subgroup?

Determining if a subset is a subgroup is important because it allows us to better understand the structure of a group. It also helps us identify smaller groups within a larger group and can lead to a deeper understanding of the properties of the group.

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