Math Is Fun Forum

parthenos

Member

Registered: 2008-08-25

Posts: 9

Abstract Algebra - groups

1.  Find the order of (1 12 8 10 4)(2 13)(5 11 7)(6 9).

I am not sure what it means by order.  I am not sure if I am suppose to find the length, lcm or what.  Please help.

2.  If n is odd and n ≥ 3, show that the identity is the only element of D2n which commutes with all elements of D2n. 

3.  If τ = (1 2)(3 4)(5 6)(7 8)(9 10) determine whether there is a n-cycle σ (n ≥ 10) with τ = σ^k for some integer k.

Please help...thanks

Last edited by parthenos (2008-08-31 12:43:01)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Abstract Algebra - groups

1. Order is the *minimal* number of times an element must be multiplied (or added) by itself to get back to the identity.  If there is so such number, we say it has infinite order.  You should be familiar with the integers modulo n.  Setting n=4, we have that the order of 0, or o(0), is 1.  o(1) = 4, o(2) = 2, and o(3) = 4.

There is a special trick with permutations however.  It seems as if you should have either been given that as a theorem or you're supposed to figure it out now.  It comes from the following observation:

o( (123) ) = 3
o( (45) ) = 2
o( (67) = 2
o( (123) (45) ) = 6
o( (123) (45) (67) ) = 6

If you calculate those by hand, you should be able to see what's going on.

2. Remember the relation in all D2n groups: rs = sr-¹.  If you have never seen that before (r is a rotation, s is a flip in the plane), then I'm not certain how you would approach the problem.

---

"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 - groups

Jane... you really shouldn't be giving out full answers.  These are most likely homework assignments.  When someone asks a question with no work, hints at most should be given.  Perhaps even better is to just identify what should be known to solve the problem.

---

"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..."

cmcneil

Member

Registered: 2008-09-04

Posts: 3

Re: Abstract Algebra - groups

The class I am taking is Abstract Algebra (grad level).  The problem I have having is with Basic Axioms and Groups.  The problem states:  Let G = {a+b√2 is an element of R|a,b are elements ofQ}.  Prove that G is a group under addition.  Do I pick random rational numbers to represent a and b to prove this theorem?  I know you have to prove the identity and inverse are unique; the inverse of the inverse is the original element, and the binary operation (in this case, addition) is true under associative and commutative.

Another question; I am given the following info: Find the orders of each element Z/12Z. Now, all I know about this problem is n=12 and the class set is 0 bar through 11 bar. Should I add 12 to each class in order to find the remainder. I know the order is exponent on the number when the remainder is one.

I hope you can help me.  This is time sensitive!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Abstract Algebra - groups

Do I pick random rational numbers to represent a and b to prove this theorem?

If by random you mean arbitrary, then yes.

I know you have to prove the identity and inverse are unique; the inverse of the inverse is the original element, and the binary operation (in this case, addition) is true under associative and commutative.

You don't have to prove that inverses are unique.  That is a consequence of being a group.  All you need to show is that:

1. The set is non-empty.
2. There is an associative binary operation defined on the set.
3. This operation is closed under the operation and inverses.

The operation will not (in general) be commutative.

Another question; I am given the following info: Find the orders of each element Z/12Z.

Remember what the definition of order is.  You continually multiple an element by itself until you get back to the identity, counting the number of times you do this.  If this never happens, you say the order is infinite.

---

"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..."

cmcneil

Member

Registered: 2008-09-04

Posts: 3

Re: Abstract Algebra - groups

Prove that if {a(n)} is a sequence of real numbers and a(n) --->a as n--->∞, then √a(n)---> √a as n --->∞.

Please help!