Group (K-map)

Also called: K-map group, grouping, K-map grouping, K-map loop, group of 1s

A rectangle of 1, 2, 4, 8 or 16 adjacent K-map cells holding only 1s or don't-cares. Each group becomes one product term with the changing variables removed.

A group on a karnaugh map is a set of cells you circle together because they can be written as one product term. Every group is an implicant of the function.

A valid group:

holds only 1s and don't-cares, never a 0
has 2ᵏ cells: 1, 2, 4, 8 or 16
is a rectangle, possibly wrapped across an edge (see wrap around adjacency)

Why it works: a pair of adjacent cells differ in one variable, so = removes that variable. A group of 4 is two adjacent pairs, which merge again and remove a second variable. So a group of 2ᵏ cells removes k variables. In an n-variable map its term has n − k literals.

To read a group, compare the row and column labels it covers. A variable that stays 0 across the whole group appears primed; one that stays 1 appears plain; one that changes is dropped.

Good groups are as large as possible, which makes them prime implicants. A minimal answer uses as few of them as possible. Groups may overlap.

AB\CD	00	01	11	10
00	0m0	0m1	1m3	1m2
01	0m4	0m5	0m7	0m6
11	0m12	0m13	0m15	0m14
10	0m8	0m9	1m11	1m10

Worked examples

Example

Reading the group above

The group m3, m2, m11, m10 wraps between the top and bottom rows.

1.
Rows covered: AB = 00 and AB = 10. A changes; B stays 0 → keep .
2.
Columns covered: CD = 11 and CD = 10. C stays 1 → keep C; D changes.
3.
Term: . Four cells in a 4-variable map leave 4 − 2 = 2 literals. ✓

Example

A square in the middle rows

Read the group m4, m5, m12, m13.

AB\CD	00	01	11	10
00	0m0	0m1	0m3	0m2
01	1m4	1m5	0m7	0m6
11	1m12	1m13	0m15	0m14
10	0m8	0m9	0m11	0m10

1.
Codes: 0100, 0101, 1100, 1101.
2.
A changes, B stays 1, C stays 0, D changes.
3.
Term: .

Common mistakes

Circling 3, 5 or 6 cells. Group sizes must be powers of two; cover such shapes with overlapping groups.
Including a 0 to make a nicer shape. A group with a 0 in it would make the function 1 where it should be 0.
Circling an L-shape or a diagonal. Groups must be rectangles.