# Generalizing the definition of a subgroup

• I
• Mr Davis 97
In summary, the conversation discusses two subgroups of a group ##G##: ##H_a## and ##H_S##. While ##H_a## only requires its elements to commute with a single fixed element ##a##, ##H_S## requires its elements to commute with all elements of a subset ##S## of ##G##. This means that proving the former differs from proving the latter, although they may have some similarities. If ##S## is the entire group ##G##, then ##H_S## is referred to as the 'centre' of ##G##.
Mr Davis 97
Let ##G## be a group. I have shown that ##H_a = \{x \in G | xa=ax \}## is a subgroup of G, where ##a## is one fixed element of ##G##. I am now asked to show that ##H_S = \{x \in G ~| ~xs=sx,~ \forall s \in S\}## is a subgroup of ##G##. How would proving the former differ from proving the latter? Couldn't I essentially use the same proof as the former but use ##s## instead of ##a##?

The proof is not the same, because the elements of ##H_a## only need to commute with the single element ##a##, whereas the elements of ##H_S## needs to commute with all elements of ##S##, which we assume (although it is not stated) is a subset (not necessarily a subgroup) of ##G##. If ##S## has only one element, the proofs will be the same. Otherwise not, although they may be similar.

By the way, if ##S=G## then ##H_S## is called the 'centre' of the group ##G##.

## What is the definition of a subgroup?

The definition of a subgroup is a subset of a group that still maintains the group's operations and structure, making it a smaller version of the original group.

## How can the definition of a subgroup be generalized?

The definition of a subgroup can be generalized by expanding it to apply to a wider range of mathematical objects, such as rings, fields, and vector spaces.

## What are the properties of a subgroup?

A subgroup must have the same identity element as the original group, and its elements must be closed under the group's operations. It must also contain the inverse of every element in the subgroup.

## What is the significance of generalizing the definition of a subgroup?

Generalizing the definition of a subgroup allows for a deeper understanding of the structures and relationships between different mathematical objects. It also allows for the study of substructures in a wider range of mathematical contexts.

## How does the concept of a subgroup relate to other mathematical concepts?

The concept of a subgroup is closely related to other mathematical concepts such as cosets, quotient groups, and normal subgroups. It also plays a crucial role in group theory, abstract algebra, and other branches of mathematics.

• Linear and Abstract Algebra
Replies
11
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
15
Views
2K
• Linear and Abstract Algebra
Replies
5
Views
1K
• Linear and Abstract Algebra
Replies
18
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
2K
• Linear and Abstract Algebra
Replies
3
Views
2K
• Linear and Abstract Algebra
Replies
6
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
2K