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loida

Member

Registered: 2009-11-12

Posts: 32

abstract algebra problem(groups, rings)

i need help with few exercises.. if there is anyone willing to give a look, I will be grateful

1)prove that every finite subgroup of Q/Z is cyclic.
2)how many sylow 7-subgroups are in A8?  ( i think 8, but i'm not sure how to prove it)
3)let R={a/b; a,b ∈ Z and b is not divisable by 11} Show that R is factorization domain. How many prime elements are in R up to associates (Elements a,b are said to be associates if a | b and b | a. ) ? How do ideals look like in R?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: abstract algebra problem(groups, rings)

Hint for #1:

Let

be a finite subgroup of

where

and

. Set

. Now show that

.

Last edited by JaneFairfax (2010-01-07 11:23:12)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: abstract algebra problem(groups, rings)

#2: Using trivial Sylow theory, post the list of possibilities and we'll help you from there.

#3: Prove that irreducibles are primes and then show that every element can be written as a product of irreducibles and units.  What do you think irreducible elements look like in this ring?  Prove it.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

loida

Member

Registered: 2009-11-12

Posts: 32

Re: abstract algebra problem(groups, rings)

wow! thank u veery much Jane&Ricky! but my knowledge from this is really poor

Hint for #1:

Now show that

.

(blushing)  i only know (a)={a^n| n∈Z } ..could u pls show me how?

Ricky...about #2, i've been reading ..so, using 3rd Sy theorem (k*7+1 | |A8|)  |A8|=20160  so..possibilities are for k=0,1,2,5,9,17,41,137 (i think so)

Last edited by loida (2010-01-08 09:39:00)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: abstract algebra problem(groups, rings)

(It's very important you understand how this is wayyy more meaningful than 20160 )

Now Sylow's theorem says that if n_7 is the number of Sylow 7-subgroups, then

The RHS restriction will give you 2^9 possible numbers right off the bat, and the LHS should reduce this into a manageable amount.  I'm not sure how do to do this one without aid of a computer, but the list of possibilities is:

1, 8, 15, 36, 64, 120, 288, 960

Now 1 is out immediately (why?).  The goal is to try to eliminate all but one.  This can be tricky.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: abstract algebra problem(groups, rings)

After giving it a few moments thought, I think you must have the problem wrong.  There are techniques for dealing with Sylow theory for groups of medium order, but they are advanced and never covered in an introductory class.  Given the difficulty level of the other problems you posted, I'm thinking something is wrong.  Perhaps I'm missing something, but that's my best guess.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

loida

Member

Registered: 2009-11-12

Posts: 32

Re: abstract algebra problem(groups, rings)

Ricky..thanks alot!!! i really appreciate ur help

so, 1 is out cause p=7 prime, |G|>1
can i say that A_8 has (8*7*6*5*4*3*2)/7 cycles of length 7, which means 5760 cycles 7-cycle
i get 960 subgroups of order 7 from this cycles..960=1(mod7) and 960| |A_6| ..??

#3 ..dont know

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: abstract algebra problem(groups, rings)

so, 1 is out cause p=7 prime, |G|>1

No.  1 is out because A_8 is a simple group, and if there is only 1 Sylow 7-subgroup, then this subgroup must necessarily be normal.

can i say that A_8 has...

Yes, this works.  You must prove that each element of order 7 in A_8 is indeed a 7-cycle, and then counting them suffices.  No Sylow theory needed at all.

For #3: I'll give you a hint:  Certainly any element a/b where 11 does not divide a is a unit in this ring (prove it!).  So to find irreducible, it suffices to look at elements of the form:

Now there is a restriction here, in other words it isn't true for all k and b that the number above is irreducible.  Find this restriction, and then prove all the elements you've found are also prime.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

loida

Member

Registered: 2009-11-12

Posts: 32

Re: abstract algebra problem(groups, rings)

tnx Ricky