You are Here: Home >< Maths

# A subgroup of a group ... Watch

1. Hi everyone,

In the case where a set A is finite included in a group G and that we show that 1 is in it and that it is closed under multiplication (the operation of G), then I get that A is a subgroup of G.

Is this correct ?

We don't need to show that the inverse of an element of A is in G, because if a is in A then there exists n such that a^n is equal to 1 then a^(n-1) is the inverse of a. (all we use here is the finiteness of A and the fact that it is close under multiplication.

What do you think ?
2. That's fine (as long as you can prove that if then ).
3. (Original post by nuodai)
(as long as you can prove that if then ).
Well, we can get away with proving something weaker: such that - which is arguably less work.
4. (Original post by Unbounded)
Well, we can get away with proving something weaker: such that - which is arguably less work.
Good point!
5. (Original post by Unbounded)
Well, we can get away with proving something weaker: such that - which is arguably less work.

(Original post by nuodai)
Good point!

Well my point was that what you say doesn't have tobe proved since it is implied by finiteness and the closure, no matter what the group nor the set A are.
Do you agree ?
6. (Original post by hitheuk)
Well my point was that what you say doesn't have tobe proved since it is implied by finiteness and the closure, no matter what the group nor the set A are.
Do you agree ?
It depends what you mean by "have to be proved". Everything in maths has to be proved, but once a result has been proved to be true, you can assume the result is true for all subsequent work. As such, you need to know what you're allowed to assume (and "what you're allowed to assume" is usually "what appears in your notes"). If you've proved in class/lectures/whatever that each element of a finite group has finite order, then you can just quote the result without any further ado. If you haven't, then you should prove it in your working.

Assuming you're allowed to assume this result, showing closure under multiplication is sufficient to show that a set is a subgroup without needing to show that an element's inverse is contained in the subgroup.
7. (Original post by nuodai)
It depends what you mean by "have to be proved". Everything in maths has to be proved, but once a result has been proved to be true, you can assume the result is true for all subsequent work. As such, you need to know what you're allowed to assume (and "what you're allowed to assume" is usually "what appears in your notes"). If you've proved in class/lectures/whatever that each element of a finite group has finite order, then you can just quote the result without any further ado. If you haven't, then you should prove it in your working.

Assuming you're allowed to assume this result, showing closure under multiplication is sufficient to show that a set is a subgroup without needing to show that an element's inverse is contained in the subgroup.
Yeah I agree. Ta. +1.
8. This is the easiest way to check that something is a subgroup:
1. Check that it is non-empty.
2. Check that for every pair x, y of elements, is in the subgroup. This automatically takes care of inverses, identity, closure, etc.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: December 17, 2010
Today on TSR

Find out how.

### Homophobic parents forcing me far away

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.