Group theory 12: Cauchy's theorem

[@3Max]

This is such a fun approach to group theory -- great balance between theory and examples.

[@dibeos]

I really do not understand in the beginning of the proof when you say: “if p divides the order of any smaller group…”. Smaller than what? G has order smaller than what? Than p it cannot be because p cannot divide a number smaller than itself… I’m confused about this phrase. Could you please clarify?

[@ducthangnguyen0108]

Since he is using induction on the order of n, he must assume that the statement holds true for groups of order less than n.

[@robshaw2639]

why do we know that we can pick this element q of some prime order?

[@98danielray]

if you suppose an element a has composite order qc then a^c has order q.

[@robshaw2639]

@@98danielray thanx.

[@vladislavbalakirev5826]

You never explicitly state the base of induction...

[@ai314159]

I noticed that too. The smallest group whose order is divisible by p has order p and is therefore cyclic, so by definition has an element of order p.

[@kaa1el960]

complete induction does not need a base case.😀