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Product of a Finite Abelian Group [06/28/2004]
Suppose G = {a1, a2, ... , an} is a finite Abelian group. If G has odd order, what can you say about the 'product,' a1*a2*...*an, of all the elements of G? What can you say about this 'product' if G has even order? What if G is not Abelian?

Product of Disjoint Cycles [10/16/1998]
How to express (1 2 3 5 7)(2 4 7 6) as the product of disjoint cycles.

Proof of Division Algorithm [11/13/1997]
a,b are positive integars, b does not equal 0; there are unique integers q and r such that a = qb+r; 0 is less than or equal to r, r is less than modulus value of b.

Proof of Normal Sylow p-Subgroup [10/28/2005]
Let G be a finite group in which (ab)^p = a^p*b^p for every a,b in G, where p is a prime dividing O(G). Prove that the Sylow p-subgroup of G is normal.

Proof of One Step Subgroup Test [05/13/2002]
Prove that a nonempty subset H of a group G is a subgroup of G if and only if a*b^(-1) is in H for all a, b in H.

Proof of Only One Identity Properity for Binary Operations [10/31/2001]
I am trying to prove that there is one and only one identity property for every operation.

Proof of Subgroup Involving an Isomorphism [11/04/2004]
Suppose is an isomorphism from a group G to a group G'. Prove that if K is a subgroup of G, then (K) = {(k) | k is in K} is a subgroup of G'.

Proof of the Partial Fractions Theorem for Quadratic Factors [10/14/2001]
Why is it that when you have a non-reducible quadratic factor, you have to let the numerator of the partial fraction be Ax+B?

Proofs on the Order of Group Elements [10/19/1998]
Can you help me with these proofs about the order of an element? Let a be any element of finite order of a group G...

Proof That a Group is Abelian [10/29/2004]
If G is a finite group whose order is not divisible by 3, and (ab)^3 = a^3b^3 for all a,b in G, prove that G must be Abelian.

Proof that f(K) is a Subgroup of G' [11/26/2001]
If G and G' are groups, f is an isomorphism from G into G', and K is a subgroup of G, then show that the set f(K)={f(k)\k is a member of K} is a subgroup of G'.

Proof That G Is Abelian [03/05/2003]
Let H be a subgroup of G that is different from G and let x*y=y*x for all x and y in G minus H.

Properties? Axioms? What to Call Characteristics of Field, and When [07/14/2014]
An adult finds inconsistent labels for the characteristics of fields and rings. Doctor Peterson explains how different levels of abstraction warrant different vocabulary.

Prove G is a Cyclic Group [02/27/2003]
Let group G be finite Abelian such that G has the property that for each positive integer n the set {x in G such that x^n = identity} has at most n elements. Prove G is a cyclic group.

Prove Sylow-p Subgroups Abelian [05/15/2003]
G is a finite simple group with exact 2p + 1 sylow-p subgroups. Prove that each of these sylow-p subgroups is Abelian.

Proving an Element is in a Group [04/12/2002]
If G is a group, prove that the only element g in G with g^2 = g is 1.

Proving That a Symmetric Group on a Finite Set Is Not Cyclic If the Set Has More Than Two Elements [01/11/2010]
Proof of a symmetric group on a finite set not being cyclic if the set has more than 2 elements. Doctor Carter debugs the work of a novice algebraist's approach, which relies on non-commutative n-cycles; and provides some pointers on conventions of notation.

Proving That Z_{mn} is Isomorphic to Z_m X Z_n [04/22/2009]
If m and n are relatively prime, show that Zmn is isomorphic to Zm X Zn.

Quadratic Diophantine Equation [01/16/2009]
Find all positive integers N such that 2*N^2 - 2*N + 1 is the square of an odd integer.

Quadratic Fields and the Division Algorithm [08/12/2003]
How can I prove that there does not exist a division algorithm in any quadratic field K = Q(sqrt(D)), where D <= -15?

Quaternary Numbers [6/28/1995]
How do quaternary numbers work? As I understand it, its q = r + a.i + b.j + c.k where r is real and i^2 = j^2 = k^2 = -1, but what happens when you start multiplying and dividing i, j and k?

Quaternion Numbers [01/23/1997]
How do you divide quaternion numbers? Can quaternion math be extended to transcendental functions?

Quaternion Numbers in Quantum Physics [04/11/2001]
Are there any applications of quaternion numbers in quantum physics?

Questions in Modern Algebra [2/3/1996]
I am studying Modern Algebra (grad level course) and would appreciate your help in answering three questions...

Rainbow Logic [1/13/1995]
In the Family Math book, there is an activity called Rainbow Logic. It's like battleship in that one player sets up a matrix and the other player(s) have to match their matrices by asking questions...

Real Numbers Closed under Division [09/24/2002]
Is it true that real numbers are not closed under division because we can't divide by 0?

Resultant of Two Polynomials [09/13/2001]
Can you give me an example of how to find the resultant of two polynomials?

Rings and Ideals [11/13/1998]
Can you help me with the following proofs on rings, ideals, and polynomials? If R is a commutative ring, prove R[x]/(x) = R. ...

Ring Theory - a Telescoping Problem [01/20/2002]
An element a in a ring R is said to be nilpotent if there exists a positive integer n such that a^n = 0. Show that if a is nilpotent, a-1 is a unit.

Roots of Cubic Equations [7/9/1995]
How many real roots does a third degree equation in the form ax^3 + bx^2 + cx + d = 0 have, and what are they?

Roots of the Cubic Equation in F2^M [07/20/2000]
Is there a general solution for the cubic equation where x is an element of the finite field F2^M?

Show a Subset [02/09/2003]
If the number of elements in a finite group G with identity e is even...

Show That G is a Group [02/28/2002]
Let G be a finite group. Show that there exists a positive integer "m" such that a^m = e for all a in G. Suppose that G is a set closed under an associative operation such that: for every a,y in G, there exists an x in G such that ax = y; and for every a,w in G, there exsits a u in G such that ua = w. Show that G is a group.

Significance of Rational Numbers [01/11/2003]
Why are rational numbers defined the way they are?

Solvable Groups [12/10/2002]
What is the connection between the solvability of polynomials for degree>=5 and the solvability of An for n>=5? What is the proof that shows An is simple for n>=5?

Solving Nonlinear Systems with Many Variables [10/25/2000]
Do you know a numeric algorithm that solves systems of non-linear polynomial equations with up to 10 variables?

Special Unitary Groups In Physics [11/09/1998]
Could you explain the groups SU(2) and SU(3)? They are central to descriptions of quantum chromodynamics.

Splitting Fields of Quartic Polynomials [05/14/2004]
I picked the irreducible polynomial x^4 - 8x^2 + 8 and I tried to find its splitting field (E) and G(E/Q) set, but I got stuck after finding the roots. Could you please help me out? Thanks in advance!

Subgroup and Order of a Group [11/06/2003]
If o(G) = p^n, p is a prime number, and H is not equal to G and it is a subgroup of G, show that there exists an x that is an element of G and x is not an element of H such that x^(-1)Hx = H.

Subgroups of the Rational Numbers Under Addition [02/01/2003]
I need to describe all the subgroups of the rational numbers under addition.

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