Cayley Table & Group Calculator
Build the Cayley table of a cyclic group, the units U(n), a dihedral, symmetric or alternating group, V₄, Q₈ or a product of two cyclic groups, or paste your own table to test the group axioms. Get the identity, inverses, element orders, generators, centre, conjugacy classes, subgroups and isomorphism type, with every step shown.
How to Use
- Pick a group: Zn (addition mod n), U(n) (multiplication mod n), Dn (symmetries of an n-gon), Sn or An (permutations), V₄, Q₈ or Zm × Zn.
- Enter n (and m for Zm × Zn). The table and every result update as you type.
- Or choose Custom and paste a table: n rows of n symbols, optionally with a heading row that starts with
*and a heading column. Without headings, whole numbers are read in increasing order and other symbols in the order they first appear. The checker tests closure, associativity, identity and inverses and names the first thing that fails. - Or press a preset, such as U(10), D₄, Q₈ or the multiplication-mod-10 table that is not a group.
- Click any heading or cell of the colour-coded table to highlight its row and column and every place its value appears. The panel shows that element’s order and the cycle of its powers.
- Read the elements, conjugacy classes and subgroups under the table and the reasoning in Show Work, and copy the table as CSV or LaTeX.
Cayley Table
The table appears here.
Worked Example
U(10), the units mod 10. The elements are the numbers from 1 to 9 that share no factor with 10: {1, 3, 7, 9}, so |U(10)| = φ(10) = 10 × (1 − 1/2) × (1 − 1/5) = 4. Multiplying mod 10 gives the table:
| · | 1 | 3 | 7 | 9 |
|---|---|---|---|---|
| 1 | 1 | 3 | 7 | 9 |
| 3 | 3 | 9 | 1 | 7 |
| 7 | 7 | 1 | 9 | 3 |
| 9 | 9 | 7 | 3 | 1 |
The powers of 3 are 3, 9, 27 ≡ 7 and 21 ≡ 1, so 3 has order 4 = |U(10)| and generates the whole group. 7 also has order 4 (7, 49 ≡ 9, 63 ≡ 3, 21 ≡ 1), and 9 has order 2 because 81 ≡ 1. The orders are 1, 4, 4, 2: U(10) is cyclic, generated by 3 and 7, and U(10) ≅ Z₄. The inverses can be read off the table: 3⁻¹ = 7 and 9⁻¹ = 9.
Common mistake 1: putting non-units in U(n). 0, 2, 4, 5, 6 and 8 are not in U(10). 2 × x mod 10 is always even, so it is never 1 and 2 has no inverse; 0 times anything is 0. Writing out the whole multiplication table mod 10 (the “× mod 10” preset) gives a closed, associative operation with identity 1, but those six elements have no inverses, so it is not a group.
Common mistake 2: assuming every group of order 4 is cyclic. U(8) = {1, 3, 5, 7} also has 4 elements, but 3² = 9 ≡ 1, 5² = 25 ≡ 1 and 7² = 49 ≡ 1 (mod 8). No element has order 4, so U(8) is not cyclic: it is the Klein four-group V₄ ≅ Z₂ × Z₂, with 5 subgroups against the 3 of Z₄.
Show Work
Formulas
Background: From Cayley’s Tables to Burnside’s Question
Cayley’s 1854 paper. Arthur Cayley introduced the multiplication table in “On the theory of groups, as depending on the symbolic equation θn = 1”, published in the Philosophical Magazine (4th series, vol. 7, pp. 40–47) in 1854. At a time when the only groups studied were groups of permutations, he defined a group abstractly, by symbols and how they combine, and displayed the combinations as a table. He also wrote that the symbol xy means performing y first and then x, and that x(yz) = (xy)z. This calculator uses the same convention, so in S₃ the product (12)∘(13) applies (13) first.
Lagrange’s theorem. Joseph-Louis Lagrange did not state the theorem in its modern form. In his Réflexions sur la résolution algébrique des équations (Berlin, 1771) he observed that permuting the n variables of a polynomial in all n! ways gives a number of different polynomials that divides n!. Gauss proved the case of the non-zero integers mod a prime in 1801, Cauchy the case of the symmetric group in 1844, and Camille Jordan the case of any permutation group in 1861. The converse is false: A₄ has 12 elements and no subgroup of order 6, which this calculator reports in Show Work.
How many groups of each order? Up to isomorphism there are 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1 and 14 groups of orders 1 to 16 (OEIS A000001), and 2,328 of order 128. A prime order allows only the cyclic group. This calculator names every abelian group (from how many elements have each order), every dihedral group (from its defining relations) and every group of order up to 15; it does not guess beyond what it can prove.
Element orders and Burnside’s question. In a finite group every element has finite order, dividing |G|. In 1902 William Burnside asked whether the converse holds for groups with finitely many generators: if every element has finite order, must the group be finite? Evgeny Golod and Igor Shafarevich showed in 1964 that it need not be, with an infinite p-group generated by three elements: every element has finite order, a power of p, but there is no uniform bound on the orders. A stricter version asks the same of a finitely presented group, one with finitely many generators and finitely many relations. On 23 September 2026 OpenAI published an AI-generated preprint, “An infinite finitely presented periodic group”, that claims a counterexample: a Steinberg group over an algebra of characteristic two. Its Lean scope note for result #247 says the formalization constructs an infinite finitely presented periodic group, where “periodic” means each element has some finite order with no common exponent, and it links a machine-checked Lean statement published with Lean Comparator checks. The claim has not yet been peer reviewed. What such a check does and does not establish, with #247 as the worked example, is explained in what “Lean-checked” means.
Sources
- A. Cayley, On the theory of groups, as depending on the symbolic equation θn = 1, Philosophical Magazine (4th series) 7 (1854) 40–47, doi:10.1080/14786445408647421; and the MacTutor biography of Arthur Cayley (his definition, the tables and the order of composition).
- OEIS A000001, number of groups of order n (values from its b-file).
- Wikipedia, Lagrange’s theorem (group theory), History section, citing Lagrange (1771), Gauss (1801), Cauchy (1844) and Jordan (1861).
- W. Burnside, On an unsettled question in the theory of discontinuous groups, Quarterly Journal of Pure and Applied Mathematics 33 (1902) 230–238.
- E. S. Golod, On nil-algebras and finitely approximable p-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 273–276; with I. R. Shafarevich, On the class field tower, same volume, 261–272. Summarised in Wikipedia’s Burnside problem and Golod–Shafarevich theorem articles.
- OpenAI, Lean scope note for result #247 and the preprint An infinite finitely presented periodic group (23 September 2026; AI-generated, not peer reviewed).
About This Calculator
This calculator builds the Cayley table of a finite group and works out its structure: the identity, every inverse and element order, whether it is abelian or cyclic and which elements generate it, the centre, the conjugacy classes, every subgroup and which of them are normal, and its isomorphism type where that can be proved. It handles Zn, U(n), Dn, Sn and An up to n = 5, V₄, Q₈ and Zm × Zn, for groups of up to 200 elements, and draws the table up to 60.
A pasted table is not assumed to be a group. The checker tests closure, all n³ triples for associativity, the identity and every inverse, and names the first triple or element that fails, so you can see exactly why a table is or is not a group. The subgroup search joins cyclic subgroups until nothing new appears; if a group would need more than 1,500 subgroups or 10 million products, it lists the cyclic ones and says so. All calculations run in your browser; nothing you enter is sent to a server.
It is meant for abstract algebra students checking homework, teachers preparing examples, and puzzle and cryptography hobbyists looking for generators and primitive roots.
Related calculators: Finite Field Calculator, Permutation and Combination Calculator, and Prime Factorization Calculator.
Frequently Asked Questions
How do I make the Cayley table of U(10)?
List the units first: U(10) = {1, 3, 7, 9}, the numbers below 10 that share no factor with 10, so φ(10) = 4. Then multiply mod 10: 3·3 = 9, 3·7 = 21 ≡ 1, 3·9 = 27 ≡ 7, 7·7 = 49 ≡ 9, 7·9 = 63 ≡ 3 and 9·9 = 81 ≡ 1. The orders are 1, 4, 4 and 2, so U(10) is cyclic, generated by 3 or 7, and isomorphic to Z₄.
How can I tell whether a table is a group?
Check four things: every entry is an element (closure), (ab)c = a(bc) for every triple, there is an identity, and every element has an inverse. Being a Latin square is necessary but not enough. The order-5 loop preset is a Latin square with identity 0 in which every element is its own inverse, yet (1∗1)∗2 = 0∗2 = 2 while 1∗(1∗2) = 1∗3 = 4, so it is not associative.
Is every group of order 4 cyclic?
No. There are two groups of order 4: Z₄, with element orders 1, 2, 4, 4, and the Klein four-group V₄ ≅ Z₂ × Z₂, with orders 1, 2, 2, 2. U(8) = {1, 3, 5, 7} is a Klein four-group: 3² = 9 ≡ 1, 5² = 25 ≡ 1 and 7² = 49 ≡ 1 (mod 8), so no element has order 4.
What does Lagrange’s theorem say about element orders?
The order of every subgroup divides the order of the group, and so does the order of every element, because g generates a subgroup of size ord(g). In D₄, of order 8, one element has order 1, five have order 2 and two have order 4. The converse is false: A₄ has 12 elements but no subgroup of order 6.
What is the difference between D₄ and Q₈?
Both are non-abelian of order 8 with a centre of size 2 and 5 conjugacy classes (8 = 1 + 1 + 2 + 2 + 2). The element orders tell them apart: D₄ has five elements of order 2, Q₈ has only one (−1) and six of order 4. D₄ has 10 subgroups; Q₈ has 6, and every one of them is normal.
How do I use the Cayley Table & Group Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Abstract algebra homework
Find the order of every element of D₄: r and r³ have order 4, while r² and the four reflections have order 2, giving counts of 1, 5 and 2 that all divide 8.
Primitive roots for crypto examples
U(11) is cyclic of order 10, generated by 2, 6, 7 and 8, the primitive roots mod 11. U(15) has 8 elements but is not cyclic (it is Z₂ × Z₄), so 15 has no primitive root.
Same order, different group
Z₈ has 4 generators (1, 3, 5, 7). Z₂ × Z₄ also has 8 elements but none of order 8: three of order 2 and four of order 4.
Permutation practice
In S₃, (12)∘(13) = (132) but (13)∘(12) = (123), so S₃ is not abelian. Its 6 elements are the identity, three transpositions and two 3-cycles.
Checking a table you wrote
Multiplication mod 10 on {0, …, 9} is closed and associative with identity 1, but 0, 2, 4, 5, 6 and 8 have no inverse, so it is not a group. The units {1, 3, 7, 9} are.
Subgroup lattices
S₄ has 30 subgroups: 1 of order 1, 9 of order 2, 4 of order 3, 7 of order 4, 4 of order 6, 3 of order 8, and A₄ and S₄ itself.
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