Abstract algebra

• Why Do $xy$ and $yx$ Have the Same Order in a Group?
  In any group, xy and yx are conjugate: x⁻¹(xy)x = yx. Conjugate elements have the same order, finite or infinite.
• How to Prove x(y − z) = xy − xz in a Ring
  In any ring, left distributivity and the additive inverse law give x(y − z) = xy − xz; follow the proof and a worked matrix example.
• What Are the Group Tables for a Group of Four Elements?
  A group with four elements is isomorphic to either the cyclic group C4 or the Klein four-group V4; see both multiplication tables and why.
• Does the Set of Real Numbers Form a Group?
  The real numbers form a group under addition, but not under multiplication; the nonzero real numbers form a multiplicative group.
• What Are Groupification and Perfection of a Commutative Monoid?
  Groupification embeds a commutative monoid into a universal group when cancellation holds; p-perfection adjoins compatible p-th roots via Frobenius.
• Why Is Every Finite Integral Domain a Field?
  In a finite integral domain, multiplication by any nonzero element is injective and therefore surjective, which guarantees a multiplicative inverse.
• What Are Examples of Infinite Boolean Rings?
  Examples of infinite Boolean rings include power sets under symmetric difference and intersection, finite-cofinite sets, and products of copies of F₂.
• How Are Cyclic Subgroups and Their Generators Related?
  Learn how to find the generators and order of a cyclic subgroup, and why each divisor of a finite cyclic group's order gives one subgroup.
• Why Is Every Subgroup Containing the Commutator Subgroup Normal?
  If H contains the commutator subgroup [G,G], every conjugate of an element of H differs from it by a commutator, so H is normal.
• How to Make Addition and Multiplication Tables for a Ring or Ideal
  Build ring and ideal operation tables by applying the inherited operations to each pair; see a complete example using Z4 and its ideal {0, 2}.
• Direct Sum vs. Direct Product vs. Tensor Product
  A direct sum restricts infinite tuples to finite support, a direct product allows all tuples, and a tensor product combines spaces bilinearly.
• Find All Irreducible Polynomials of Degrees 2 and 3 in F₂[x]
  The irreducible polynomials in F₂[x] of degrees 2 and 3 are x²+x+1, x³+x+1, and x³+x²+1; a root test proves the list.
• Does Having No Roots Make a Degree-2 or Degree-3 Polynomial Irreducible?
  For polynomials of degree 2 or 3, rootlessness implies irreducibility over a field, but not over every integral domain; learn the needed conditions.
• Is the Tensor Product of a Module Element and Zero Always Zero?
  In a tensor product of modules, m ⊗ 0 and 0 ⊗ n are always zero. Bilinearity proves this, but sums of pure tensors need separate analysis.
• What Is a Quotient Ring of the Gaussian Integers?
  A quotient of the Gaussian integers depends on the ideal: learn how to form Z[i]/(α), count its elements, and determine when it is a field.
• How to Test Whether an Element of a Tensor Product Is Zero
  To test whether a tensor is zero, use a module presentation or a separating map; pure tensors and vector-space tensors have useful special criteria.