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Browse College Modern Algebra
Stars indicate particularly interesting answers or
good places to begin browsing.
 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 pSubgroup [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 psubgroup 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 ö is an isomorphism from a group G to a group G'. Prove that
if K is a subgroup of G, then ö(K) = {ö(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 nonreducible 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 Sylowp Subgroups Abelian [05/15/2003]

G is a finite simple group with exact 2p + 1 sylowp subgroups. Prove
that each of these sylowp 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 noncommutative ncycles; 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, a1 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 nonlinear
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.
 Sylow PSubgroups of Symmetric Groups [05/13/2009]

Let p be an odd prime. First, find a set of generators for a pSylow
subgroup K of S_p^2 (the symmetric group with degree p^2). Then find
the order of K and determine whether it is normal in S_p^2 and if it
is Abelian.
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