# Abstract Algebra Help

Discussion in 'Community Discussion' started by Shaun.P, Mar 12, 2009.

1. ### Shaun.P macrumors 68000

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Jul 14, 2003
Location:
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#1
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.

2. ### RHutch macrumors 6502

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May 21, 2003
Location:
Amsterdam, OH
#2
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.

3. ### Shaun.P thread starter macrumors 68000

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Location:
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#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.

4. ### acearchie macrumors 68040

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Jan 15, 2006
#4
So is this a form of statistics or something completely different altogether?

5. ### dukebound85 macrumors P6

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#5
i covered it when i took "linear algebra and matrices" in college

and yea, i forgot how to do that crap lol

6. ### Shaun.P thread starter macrumors 68000

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Location:
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#6
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

7. ### jecapaga macrumors 601

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8. ### Frisco macrumors 68020

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Sep 24, 2002
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Utopia
#8
Wow Abstract Algebra is definitely a few steps above Calculus, which I was lucky to pass.

9. ### emmawu macrumors 6502

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Jan 19, 2005
Location:
Wauwatosa, WI
#9
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).

10. ### RHutch macrumors 6502

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Location:
Amsterdam, OH
#10
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.

11. ### magamo macrumors 6502

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Apr 6, 2009
#11
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.

12. ### RHutch macrumors 6502

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May 21, 2003
Location:
Amsterdam, OH
#12
I'm confused; are you saying that I should not have commented on the powers?

13. ### magamo macrumors 6502

Joined:
Apr 6, 2009
#13
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.

14. ### P-Worm macrumors 68020

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Jul 16, 2002
Location:
Salt Lake City, UT
#14
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

15. ### Shaun.P thread starter macrumors 68000

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#15
Abstract algebra is the name of the branch of maths this is.

16. ### jecapaga macrumors 601

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Southern California
#16
We will have to wait for his eventual arrival.

17. ### emmawu macrumors 6502

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Jan 19, 2005
Location:
Wauwatosa, WI
#17
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?

18. ### P-Worm macrumors 68020

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Jul 16, 2002
Location:
Salt Lake City, UT
#18
I know. I guess the joke didn't come out so well.

P-Worm