#24660 ⟨a, b | aa=a, ababab=bb⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality

Properties

Finite structure

Elements in the center commute with all other elements.
A right zero element x satisfies yx = x for all y.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Cayley table

Idempotents are shown in bold.

1ababbab2abababb2ab3(ab)2(ba)2b3aa(ba)2b(ab)2(ba)3
11ababbab2abababb2ab3(ab)2(ba)2b3aa(ba)2b(ab)2(ba)3
aaaabababab2aba(ab)2b2ab3(ab)2a(ba)2b3aa(ba)2b2b2a
bbbab2babb2ab3(ba)2b3b3ab3b(ab)2b3ab3a(ba)3b3b3a
abababab2(ab)2b2ab3a(ba)2b3b3ab3b2b3ab3ab2ab3b3a
bababababbab(ba)2b3(ba)2b(ab)2b3ab3b(ab)2(ba)3b3a(ba)3b3b3a
b2b2b2ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
abaabaaba(ab)2(ab)2a(ba)2b3a(ba)2b2b3ab3b2b2ab3ab2ab3b3a
babbab(ba)2b3b(ab)2b3ab3(ba)3b3b3ab3b3b3ab3ab3ab3b3a
b2ab2ab2ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
b3b3b3ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
(ab)2(ab)2a(ba)2b3b2b3ab3b2ab3b3ab3b3b3ab3ab3ab3b3a
(ba)2(ba)2(ba)2b(ab)2b(ab)2(ba)3b3(ba)3b3b3ab3b3b3ab3ab3ab3b3a
b3ab3ab3ab3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
a(ba)2a(ba)2a(ba)2b2b2b2ab3b2ab3b3ab3b3b3ab3ab3ab3b3a
b(ab)2b(ab)2(ba)3b3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a
(ba)3(ba)3(ba)3b3b3b3ab3b3ab3b3ab3b3b3ab3ab3ab3b3a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a2 ⇒ a [1]
2. ab2 ⇒ b2 [3]
3. b2ab ⇒ b3 [4]
4. b4 ⇒ b3 [5]
5. (ab)3 ⇒ b2 [2]
# ab:aa=a,ababab=bb ab - -
aa=a
abb=bb
bbab=bbb
bbbb=bbb
ababab=bb

Same cardinality

20 unique, 175 total

Σ#PresentationPropertiesDescriptionφ
8628⟨a, b | bb=aa, abab=1⟩Grp 1658
91331⟨a, b | aaaa=b, bbbb=1⟩Grp Com 16ℤ1667
92051⟨a, b | aab=a, bbbb=b⟩Noc 164
103808⟨a, b | aaab=ba, abab=1⟩Grp 167
104630⟨a, b | aaaa=a, bbbb=a⟩Noc Com 16ℕ(16 = 4)1
104648⟨a, b | aaaa=b, bbbb=a⟩Noc Com 16ℕ(16 = 1)5
106205⟨a, b | aba=b, aaaabb=1⟩Grp 163
1112164⟨a, b | aaaa=aa, bbbb=a⟩Noc Com 16ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Noc Com 16ℕ(16 = 5)2
1112212⟨a, b | aaaa=bb, bbbb=a⟩Noc Com 16ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Noc 166
1113251⟨a, b | bab=aab, bbb=aa⟩Noc 16
1113259⟨a, b | bab=aba, bbb=aa⟩Noc 16
1116028⟨a, b | aaa=ab, babb=bb⟩Noc 16
1116032⟨a, b | aaa=ab, bbaa=bb⟩Noc 16
1116060⟨a, b | aaa=bb, abab=aa⟩Noc 16
1116371⟨a, b | aba=aa, bbbb=ab⟩Noc 16
1118811⟨a, b | aaa=b, abbbbb=b⟩Noc Com 16ℕ(16 = 3)2
1120251⟨a, b | aba=b, aaaa=abb⟩Noc 16
1121039⟨a, b | ab=aa, bbba=bbb⟩Noc 16