Products & Quotients

Build new groups from old ones and watch quotient groups collapse into existence

Products & Quotients

Groups can be glued together into bigger groups or squashed into smaller ones. Squashing only works along special subgroups.

Direct Product Builder

Choose the two factors to build Z_m x Z_n as a table of pairs. Each product adds the two coordinates separately.

x= group of order 6
*(0,0)(0,1)(0,2)(1,0)(1,1)(1,2)
(0,0)(0,0)(0,1)(0,2)(1,0)(1,1)(1,2)
(0,1)(0,1)(0,2)(0,0)(1,1)(1,2)(1,0)
(0,2)(0,2)(0,0)(0,1)(1,2)(1,0)(1,1)
(1,0)(1,0)(1,1)(1,2)(0,0)(0,1)(0,2)
(1,1)(1,1)(1,2)(1,0)(0,1)(0,2)(0,0)
(1,2)(1,2)(1,0)(1,1)(0,2)(0,0)(0,1)

Each cell shows (a+a' mod 2, b+b' mod 3). The grid-like structure comes from the two factors.

Go deeper:Products & Quotients

Groups can be combined to make bigger groups (direct products) or collapsed to make smaller ones (quotient groups). The direct product G x H pairs elements from two groups. A quotient group G/N glues together elements in the same coset of a normal subgroup N.

Go deeper:Direct Product Builder

Build Z_m x Z_n as a grid. Each element is a pair (a, b) and multiplication is componentwise modular addition. See how the grid structure reflects the two factors.

Key insight: Z2 x Z3 is isomorphic to Z6 (since gcd(2,3) = 1). But Z2 x Z2 is the Klein four-group V4, which is NOT cyclic -- it has no element of order 4.

Normal vs Non-Normal Subgroups

Choose a group and a subgroup, then compare its left cosets gH with its right cosets Hg. When the two lists match, the subgroup is normal.

Normal

Left Cosets (gH)

{e}
{(0 1)}
{(0 1 2)}
{(1 2)}
{(0 2)}
{(0 2 1)}

Right Cosets (Hg)

{e}
{(0 1)}
{(0 1 2)}
{(1 2)}
{(0 2)}
{(0 2 1)}

Left and right cosets are equal -- this is a normal subgroup, so we can form a quotient group.

Go deeper:Normal vs Non-Normal Subgroups

A subgroup is normal if left and right cosets are the same. Compare left cosets gH with right cosets Hg. Only normal subgroups allow quotient construction.

Key insight: In S3, the subgroup { e, (0 1 2), (0 2 1) } of rotations is normal (index 2 subgroups are always normal). But { e, (0 1) }is NOT normal -- its left and right cosets differ.

Quotient Group Collapse

Choose S3 or D4 and a normal subgroup N, then press Collapse! Each coset shrinks to a single point, leaving the quotient group G/N.

Go deeper:Quotient Group Collapse

Watch the most dramatic visualization in group theory: start with a full Cayley diagram, color by cosets of a normal subgroup, then animate the nodes collapsing to their coset centroids. The result is the quotient group.

Key insight: The quotient group G/N "forgets" the internal structure of each coset, keeping only how cosets multiply with each other. S3/A3 collapses to Z2 -- just "even or odd."

Key Takeaways

  • Direct product -- G x H combines two groups into a larger one
  • Normal subgroup -- left cosets = right cosets; required for quotients
  • Quotient group -- G/N collapses cosets into single elements
  • The collapse animation -- visually shows how quotients simplify structure

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