Compute automorphisms for cyclic groups and graphs fast. Control enumeration depth and view explicit mappings. Clear tables support proofs, homework, research notes, and reports.
| Example | Inputs | Expected highlights |
|---|---|---|
| Z8 | Structure: Zn, n=8 | φ(8)=4; multipliers a∈{1,3,5,7} |
| Cycle graph C4 | n=4, matrix of a square | 8 automorphisms; dihedral symmetries |
For Z_n, every automorphism is multiplication by a unit a modulo n, so the total equals Euler’s totient φ(n). When n factors as ∏ p_i^{e_i}, φ(n)=n∏(1−1/p_i), which explains why prime-rich n yields fewer units. For example, n=8 gives φ(8)=4, while n=9 gives φ(9)=6. If n is prime p, then φ(p)=p−1, so every nonzero residue defines a distinct automorphism.
For graphs, the calculator tests permutations π that keep adjacency invariant: A[i][j]=A[π(i)][π(j)]. This matches the formal definition of a graph automorphism and counts the symmetry group of the graph. A 4-cycle has 8 automorphisms, reflecting its dihedral structure. By contrast, a path on four vertices has only 2, corresponding to a reflection that reverses endpoints. Dense random graphs often have only the identity.
Graph enumeration grows factorially: n! permutations in the worst case. The limit parameter caps the search and the displayed list, helping you trade completeness for speed on larger inputs. Count-only mode skips listing units or permutations, reducing output overhead and focusing on the key cardinality. For cyclic groups, enumeration is linear in n, but listing all units can still be large, so the same limit keeps tables manageable.
Results are presented as mapping rules for Z_n and as one-line and cycle notation for graph permutations. These representations support checking, comparison with answers, and reuse in notes. Cycle notation highlights fixed points and orbit structure, useful when reasoning about stabilizers. CSV export provides structured rows for spreadsheets, while PDF export creates a summary suitable for assignments.
Use small n to validate theory, then scale gradually while monitoring the “checked permutations” counter. For graphs, ensure the matrix is n×n with 0/1 entries and consistent row lengths; malformed rows are flagged. If your graph is undirected, a symmetric matrix avoids unintended directed edges. Re-running with different limits lets you confirm stability of the automorphism count and document assumptions.
The tool counts units modulo n. Each unit a with gcd(a,n)=1 defines an automorphism x ↦ a·x (mod n), so the total equals φ(n).
A brute-force search may test up to n! permutations. Even small increases in n multiply runtime, so n≤8 is recommended and the enumeration limit can cap the search.
No. It skips listing maps and focuses on totals. For Z_n it still computes φ(n); for graphs it still checks permutations, just without rendering large tables.
Enter exactly n rows with n entries each, using 0 or 1. Separate entries by spaces or commas. Newlines separate rows.
One-line notation lists images of 1..n. Cycle form groups vertices moved together, which helps interpret symmetries and fixed points at a glance.
Yes. Export CSV to compare in a spreadsheet, or export PDF for a clean record. For graphs, you can also cross-check with known families like cycles, paths, and complete graphs.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.