Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question on a proof by induction

  1. Sep 30, 2012 #1
    Hello,
    I have a subgroup [itex]S=\left\langle A \right\rangle[/itex] generated by the set A, i.e. [itex]S=\left\{ a_1 a_2 \ldots a_n \;|\; a_i \in A \right\}[/itex].

    When I need to prove by induction on n some property of S, what should I choose as the base case of induction? n=1, or simply n=0 ?

    If the answer is n=0, then it seems to me that in most cases associated with "subgroups generated by some set", we always have to define n=0 as corresponding to the identity element, thus the basis of induction will be always about proving that some property holds for the identity element. Am I right?
     
  2. jcsd
  3. Oct 1, 2012 #2

    Erland

    User Avatar
    Science Advisor

    Well, I would say that this depends upon how the statement to be proved is formulated. If it says that it should hold for all n>=0, then your interpretation is correct in my opinion. If it should hold only for all n>=1, the problem does not occur.
     
  4. Oct 1, 2012 #3
    Proofs such as you've described typically involve induction on the length of the expression. For example if a group is Abelian, meaning ab = ba, you can use induction to show that you can freely permute the factors of a product of n elements.

    Without more info it's difficult to answer your question. In the most general case you can start an induction anywhere.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Question on a proof by induction
  1. Proof by Induction (Replies: 7)

  2. Induction Question (Replies: 7)

  3. Induction Proof (Replies: 4)

  4. Proof by induction (Replies: 4)

Loading...