Let $a_1,a_2,\dots, a_n\in G$. Prove $a_1\circ\dots\circ a_n$ is unique, regardless of the order $n$ in which operations are carried out.

Asked Oct 06 '21 at 11:30

Active Oct 06 '21 at 12:32

Viewed 65 times

Let $G$ be a group with binary operation $\circ$. Let $a_1,a_2,....,a_n\in G$. Prove that $a_1\circ a_2\circ...\circ a_n$ is unique, regardless of the order $n$ in which operations are carried out.

I thought associativity might be helpful. By associativity, $a_1\circ a_2\circ a_3=(a_1\circ a_2)\circ a_3=a_1\circ( a_2\circ a_3).$

I couldn't conclude something helpful.

Any help will be appreciated. Thanks!

edited Oct 06 '21 at 12:32

Shaun

44,997

asked Oct 06 '21 at 11:30

user955446

1

Hint: use induction on $n$. – Jeffrey Wang Oct 06 '21 at 11:34
Do the list of elements $a_1, \cdots, a_n$ exhaust $G$, i.e. are they all the elements of $G$? – Elchanan Solomon Oct 06 '21 at 11:34
@Jeffrey Wang: Could you show me some steps I am feeling hard to do? – user955446 Oct 06 '21 at 11:38
1

What does it mean for some element to be "unique"? Do you mean $a_1\circ a_2\circ\cdots\circ a_n$? – 5xum Oct 06 '21 at 11:51
2

@JeffreyWang, I agree with the OP that more of a hint would be helpful. This looks like it calls for a surprisingly subtle induction argument. The number of ways to parenthesize a product grows like a Catalan number, and you have to account for all of them when you induct. – Barry Cipra Oct 06 '21 at 11:52
@5xum, I'm sure that's what the OP meant. (Good catch. I didn't even notice there were missing ellipses.) – Barry Cipra Oct 06 '21 at 11:56
@5xum: I have edited. Thanks! – user955446 Oct 06 '21 at 12:03
See also https://math.stackexchange.com/q/3250060. – Paul Frost Oct 06 '21 at 12:04

Let $a_1,a_2,\dots, a_n\in G$. Prove $a_1\circ\dots\circ a_n$ is unique, regardless of the order $n$ in which operations are carried out.

0 Answers0