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Abstract Algebra Help

Shaun.P

Shaun.P

macrumors 68000
Original poster
Jul 14, 2003
1,601
24
Omicron Persei 8
Hi all,

I was wondering if any maths people could help me with a question that I am stuck at that is really bothering me. [See picture below].

I know the order of the set S15 is 15! (15 factorial).

I also know how to prove the set H is a supgroup of S15.

Can anyone tell me though, what the order of the subgroup is? Is it the same as the order of the set S15?

Any help would be greatly appreciated.
 
I have not done abstract algebra for several years, so there may be a better solution, but I think this is correct.

In this case it is easy enough to directly compute the powers of alpha. Since there are only 6 permutations of three numbers (1, 14, 15), there are no more than 6 powers of alpha.

All numbers except 1, 14, and 15 are fixed.
Consider alpha squared: (1 14 15)(1 14 15).
1 -> 14 -> 15
14 -> 15 -> 1
15 -> 1 -> 14

So (1 14 15)(1 14 15) = (1 15 14)

Then alpha cubed is (1 14 15)(1 15 14) and
1 -> 15 -> 1
15 -> 14 -> 15
14 -> 1 -> 14

So alpha cubed is the identity.

So H = {alpha, alpha^2, and alpha^3}. The order of the subgroup H is 3.
 
I never thanked you for your help.

Thank you.

I somehow managed to unsubscribe from this thread by mistake, and never noticed there was a reply.
 
So is this a form of statistics or something completely different altogether?
 
acearchie said:
So is this a form of statistics or something completely different altogether?
Click to expand...

This is something completely different.

The actual name of the course at my Uni is Algebraic Structures.

This is the course plan:



Mappings
Composition
Binary operations
Composition as a binary operation
Examples of groups
Permutations
Subgroups
Equivalence relations
Congruence. The division algorithm
Integers modulo n
Euclidean algorithm
Definition and examples of rings
Ideals
Integral domains. Subrings
Fields
Isomorphism. Characteristic
Ordered integral domains
The integers
Field of quotients
Field of rational numbers
Extensions of fields
Definition and elementary properties of polynomials
The division algorithm
Factorisation of polynomials
Unique factorisation domains
Simple extensions
Finite fields
Partially ordered sets
Lattices
Boolean algebras
 
The order of the subgroup H must be a divisor of 15! AND equals the minimal n such that alpha^n = alpha^n+1.

Mambochicken (emmawu's so).
 
emmawu said:
The order of the subgroup H must be a divisor of 15! AND equals the minimal n such that alpha^n = alpha^n+1.

Mambochicken (emmawu's so).
Click to expand...

I think you made a typo or some other error. Don't you mean that it is the minimal n for which alpha = alpha^(n + 1)? As you have written it, you would have consecutive powers of alpha equal to each other: n and n + 1.
 
RHutch said:
I think you made a typo or some other error. Don't you mean that it is the minimal n for which alpha = alpha^(n + 1)? As you have written it, you would have consecutive powers of alpha equal to each other: n and n + 1.
Click to expand...

Well, I don't want to be an ass, but since you corrected the error, I think the order is the minimal positive integer n >= 2 s.t. alpha = alpha^(n + 1) in this case. Otherwise, a smartass might say alpha = alpha^1 and hence the order has to be 0. An even more annoying ass could argue alpha = alpha^(-2) = alpha^(-5) =... because H happened to be the prime field of order 3.
 
magamo said:
Well, I don't want to be an ass, but since you corrected the error, I think the order is the minimal positive integer n >= 2 s.t. alpha = alpha^(n + 1) in this case. Otherwise, a smartass might say alpha = alpha^1 and hence the order has to be 0. An even more annoying ass could argue alpha = alpha^(-2) = alpha^(-5) =... because H happened to be the prime field of order 3.
Click to expand...

I'm confused; are you saying that I should not have commented on the powers?
 
RHutch said:
I'm confused; are you saying that I should not have commented on the powers?
Click to expand...

No. Sorry, I expressed myself poorly... You're absolutely right about "alpha^n = alpha^n+1" is a typo or an error. I just wanted to make it complete because you kindly corrected part of emmawu's post.
 
Guys, he specifically asked for Abstract algebra help. Why are we answering the question when it is intended for him? Where is Abstract, anyway?

P-Worm
 
Hi,

Mambochicken (a mathematics professor) admitted he made a typographical error. Maybe he should be taken to the parking lot and shot, huh? Or would that be too harsh?
 
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