Group theory

• How to Find the Order of a Point on an Elliptic Curve
  The order of a point is the least positive multiple that equals the identity. Learn how to compute it over finite fields and beyond.
• 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.
• What Is the Index of a Subgroup in a Finite Group?
  The index [G:H] of a subgroup H in a finite group G is the number of cosets, equal to |G| divided by |H|.
• Does Every Infinite Abelian Group Have an Element of Infinite Order?
  No. The direct sum of infinitely many copies of the cyclic group of order 2 is infinite, abelian, and has no element of infinite order.
• Why the Map f(x) = ax Is Bijective in a Group
  For a fixed element a in a group G, left multiplication f(x)=ax is bijective because its inverse is multiplication by a⁻¹.
• Why Does (a * b)^n = a^n * b^n in an Abelian Group?
  In an abelian group, powers distribute over the operation: (a*b)^n = a^n*b^n for every integer n. See the induction proof and a nonabelian counterexample.
• 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.
• Why Must a Group of Order 27 Have a Subgroup of Order 3?
  Every group of order 27 has a subgroup of order 3: use Lagrange’s theorem to find an element whose suitable power has order 3.
• 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.
• Does a Group Generated by Two Involutions Have to Be Infinite?
  A group generated by two elements of order 2 need not be infinite or non-abelian; its structure is determined by the order of their product.