[ 3 / biz / cgl / ck / diy / fa / ic / jp / lit / sci / vr / vt ] [ index / top / reports ] [ become a patron ] [ status ]
2023-11: Warosu is now out of extended maintenance.

/sci/ - Science & Math

View post   
File: 39 KB, 374x347, a5b887d8084366a7507370f5e6ed1e71.jpg [View same] [iqdb] [saucenao] [google]
8529509 No.8529509[DELETED]  [Reply] [Original]

if you have group A = {a, b, c, e}
you know e is neutral, all members are different, it has associativity and closure.
also (a*a)*a = b

is b*a = b not possible inherently because a can't be neutral?

or is there something else that implies it's not possible?

>> No.8529519

pls help dear science tists

>> No.8529522

Why do you think it's not possible?

>> No.8529523

>>8529522
I have to prove that a*a isn't b, isn't a and isn't e

2 of these are really easy but i have no fucking idea how to prove it's not b

>> No.8529524

I like this answer. Here's another: a^4 is neutral (as the group has order 4), but a^3 = b by hypothesis. So b*a = a^4 is neutral.

Yet another way is to exhaustively list all the groups of order 4 (there are only Z/4Z and Z/2Z x Z/2Z) and match elements.

I don't think there's a direct algebraic way to just manipulate the formula.

>> No.8529527

>>8529509
Yes.
The identity is unique.
And all elements are different.
If we had b*a = B, we would also have
B*a = B*e
Since its a group we can invert which means that a = e contradicting each element being unique.

>> No.8529529

>>8529523
if a*a were b then a would equal e and you said all members are different qed

>> No.8529531

>>8529523
Well obviously (a*a) isn't b.
If it were. We would have
b*a = b which is nonsense since a is not the identity. Identity is unique in a group.

It's also not a since if it were a = e.

This is really just an exercise in know what a group is and what the axioms are. There's no creativity required here.

>> No.8529536

>>8529524
I don't think I learned this 'rule'? why is it necessary that a^4 is neutral?

I don't think I can use this as proof

>>8529527
ok so I think I get it but what confuses me is, is it really true that in no groups where theres associativity, closure, and a neutral that there isnt a single member that you can use the action on and result in the same member?

I think I can't even phrase it well, hard to translate terms from my language

I'm saying that, if there's a neutral and all members are different, is it true that for every single a in any group like this (even infinite) there's no b that a*b=b ?

>> No.8529543

>>8529531
yeah thinking about this it's very simple, my intuition told me this but i just don't know this material well

>> No.8529550

OP here unfortunately i have another question

if in a different group from the OP, e is a member of group A so that for every x in A the result of x*e=x

how do I prove that e is not "necessarily" (this wording confuses me) neutral

is it because e*e is not known so it 'could' be something other than e and it wouldn't be neutral?

this kind of answer feels weird and not mathematical

>> No.8529565

>>8529509
Whenever you see an equation in group theory just remember: every element has an inverse.

>> No.8529572

>>8529565
is that the term for a number that results in the neutral member?