Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    0
    ReputationRep:
    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 ?
    • PS Helper
    Offline

    14
    That's fine (as long as you can prove that if a \in A \le G then a^{|A|}=1).
    Offline

    12
    ReputationRep:
    (Original post by nuodai)
    (as long as you can prove that if a \in A \le G then a^{|A|}=1).
    Well, we can get away with proving something weaker:  \exists n \in \mathbb{N} such that  a^n = 1 - which is arguably less work.
    • PS Helper
    Offline

    14
    (Original post by Unbounded)
    Well, we can get away with proving something weaker:  \exists n \in \mathbb{N} such that  a^n = 1 - which is arguably less work.
    Good point!
    • Thread Starter
    Offline

    0
    ReputationRep:
    (Original post by Unbounded)
    Well, we can get away with proving something weaker:  \exists n \in \mathbb{N} such that  a^n = 1 - 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 ?
    • PS Helper
    Offline

    14
    (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.
    • Thread Starter
    Offline

    0
    ReputationRep:
    (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.
    Offline

    2
    ReputationRep:
    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, xy^{-1} is in the subgroup. This automatically takes care of inverses, identity, closure, etc.
 
 
 
  • 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
    What time of year is the worst for students?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • 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

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