#22 ⟨a, b | ab=aa, bb=1⟩

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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
  8. Isomorphic instances
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
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.

1aba2baba2
11aba2baba2
aaa2a2aaa2
bbba1ba2aa2
a2a2aaa2a2a
bababa2ba2bababa2
ba2ba2bababa2ba2ba

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. ab ⇒ a2 [1]
2. b2 ⇒ 1 [2]
3. a3 ⇒ a [3]
# ab:ab=aa,bb=1 ab - -
ab=aa
bb=1
aaa=a

Same cardinality

19 unique, 1818 total

Σ#PresentationPropertiesDescriptionφ
626⟨a, b | aaa=1, abb=1⟩Grp Com 6ℤ61373
7160⟨a, b | ab=aa, bb=b⟩Noc 64
7245⟨a, b | aa=a, bbb=a⟩Noc Com 6ℕ(6 = 3)49
7257⟨a, b | aa=b, bbb=a⟩Noc Com 6ℕ(6 = 1)61
7258⟨a, b | aa=b, bbb=b⟩Noc Com 6ℕ(6 = 2)55
8639⟨a, b | ab=aa, baa=b⟩Noc 613
8644⟨a, b | ab=aa, bbb=a⟩Noc Com 6ℕ(6 = 4)46
8893⟨a, b | aa=a, abbb=b⟩Noc 619
81011⟨a, b | ab=a, baa=bb⟩Noc 613
91427⟨a, b | abba=b, baba=1⟩Grp 666
91581⟨a, b | aaa=aa, abb=b⟩Noc 65
91648⟨a, b | aab=bb, abb=a⟩Noc Com 613
91686⟨a, b | abb=aa, bbb=a⟩Noc Com 6ℕ(6 = 5)49
93075⟨a, b | ab=a, aaaa=bb⟩Noc Com 614
93132⟨a, b | ab=a, bbbb=aa⟩Noc Com 65
93134⟨a, b | ab=a, bbbb=ba⟩Noc 62
93193⟨a, b | ab=a, bbb=aaa⟩Noc Com 69
93199⟨a, b | ab=a, bbb=bba⟩Noc 63
1124145⟨a, b | aa=a, abbbbba=b⟩Noc Com 6

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

44 total

Σ#PresentationMapping
81109⟨a, b | aa=1, aabba=b⟩φ(a) = b, φ(b) = a
81178⟨a, b | aa=1, aaba=bb⟩φ(a) = b, φ(b) = a
81181⟨a, b | aa=1, aabb=ba⟩φ(a) = b, φ(b) = a
81188⟨a, b | aa=1, abba=ab⟩φ(a) = b, φ(b) = a
81245⟨a, b | aa=1, abb=aba⟩φ(a) = b, φ(b) = a
93355⟨a, b | aa=1, aababa=b⟩φ(a) = b, φ(b) = aa
93492⟨a, b | aa=1, aabab=ba⟩φ(a) = b, φ(b) = aa
93507⟨a, b | aa=1, ababa=ab⟩φ(a) = b, φ(b) = aa
93627⟨a, b | aa=1, aaba=bab⟩φ(a) = b, φ(b) = aa
93640⟨a, b | aa=1, abab=aba⟩φ(a) = b, φ(b) = aa
109669⟨a, b | aa=1, aaaabba=b⟩φ(a) = b, φ(b) = a
109691⟨a, b | aa=1, aabaaba=b⟩φ(a) = b, φ(b) = a
109938⟨a, b | aa=1, aaaaba=bb⟩φ(a) = b, φ(b) = a
109941⟨a, b | aa=1, aaaabb=ba⟩φ(a) = b, φ(b) = a
109952⟨a, b | aa=1, aaabba=ab⟩φ(a) = b, φ(b) = a
109961⟨a, b | aa=1, aabaab=ba⟩φ(a) = b, φ(b) = a
109991⟨a, b | aa=1, abaaba=ab⟩φ(a) = b, φ(b) = a
1010212⟨a, b | aa=1, aaaab=bba⟩φ(a) = b, φ(b) = a
1010217⟨a, b | aa=1, aaaba=abb⟩φ(a) = b, φ(b) = a
1010224⟨a, b | aa=1, aaabb=aba⟩φ(a) = b, φ(b) = a
1010245⟨a, b | aa=1, aabba=aab⟩φ(a) = b, φ(b) = a
1010248⟨a, b | aa=1, aabba=baa⟩φ(a) = b, φ(b) = a
1010262⟨a, b | aa=1, abaab=aba⟩φ(a) = b, φ(b) = a
1010477⟨a, b | aa=1, aabb=aaba⟩φ(a) = b, φ(b) = a
1010487⟨a, b | aa=1, abba=aaab⟩φ(a) = b, φ(b) = a
1010499⟨a, b | aa=1, baaa=aabb⟩φ(a) = b, φ(b) = a
1010504⟨a, b | aa=1, baab=aaba⟩φ(a) = b, φ(b) = a
1126461⟨a, b | aa=1, aaaababa=b⟩φ(a) = b, φ(b) = aa
1126505⟨a, b | aa=1, aabaaaba=b⟩φ(a) = b, φ(b) = aa
1126989⟨a, b | aa=1, aaaabab=ba⟩φ(a) = b, φ(b) = aa
1127007⟨a, b | aa=1, aaababa=ab⟩φ(a) = b, φ(b) = aa
1127032⟨a, b | aa=1, aabaaab=ba⟩φ(a) = b, φ(b) = aa
1127089⟨a, b | aa=1, abaaaba=ab⟩φ(a) = b, φ(b) = aa
1127515⟨a, b | aa=1, aaaaba=bab⟩φ(a) = b, φ(b) = aa
1127536⟨a, b | aa=1, aaabab=aba⟩φ(a) = b, φ(b) = aa
1127567⟨a, b | aa=1, aababa=aab⟩φ(a) = b, φ(b) = aa
1127570⟨a, b | aa=1, aababa=baa⟩φ(a) = b, φ(b) = aa
1127614⟨a, b | aa=1, abaaab=aba⟩φ(a) = b, φ(b) = aa
1128044⟨a, b | aa=1, aaaab=baba⟩φ(a) = b, φ(b) = aa
1128055⟨a, b | aa=1, aaaba=abab⟩φ(a) = b, φ(b) = aa
1128094⟨a, b | aa=1, aabab=aaba⟩φ(a) = b, φ(b) = aa
1128100⟨a, b | aa=1, aabab=baaa⟩φ(a) = b, φ(b) = aa
1128157⟨a, b | aa=1, ababa=aaab⟩φ(a) = b, φ(b) = aa
1128226⟨a, b | aa=1, baaab=aaba⟩φ(a) = b, φ(b) = aa

Anti-isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

63 total

Σ#PresentationMapping
691⟨a, b | aa=1, abb=b⟩φ(a) = b, φ(b) = a
7328⟨a, b | aa=1, abab=b⟩φ(a) = b, φ(b) = aa
7370⟨a, b | aa=1, bab=ab⟩φ(a) = b, φ(b) = aa
81103⟨a, b | aa=1, aaabb=b⟩φ(a) = b, φ(b) = a
81113⟨a, b | aa=1, abaab=b⟩φ(a) = b, φ(b) = a
81174⟨a, b | aa=1, aaab=bb⟩φ(a) = b, φ(b) = a
81180⟨a, b | aa=1, aabb=ab⟩φ(a) = b, φ(b) = a
81195⟨a, b | aa=1, baab=ab⟩φ(a) = b, φ(b) = a
81244⟨a, b | aa=1, abb=aab⟩φ(a) = b, φ(b) = a
81247⟨a, b | aa=1, baa=abb⟩φ(a) = b, φ(b) = a
93347⟨a, b | aa=1, aaabab=b⟩φ(a) = b, φ(b) = aa
93367⟨a, b | aa=1, abaaab=b⟩φ(a) = b, φ(b) = aa
93491⟨a, b | aa=1, aabab=ab⟩φ(a) = b, φ(b) = aa
93525⟨a, b | aa=1, baaab=ab⟩φ(a) = b, φ(b) = aa
93619⟨a, b | aa=1, aaab=bab⟩φ(a) = b, φ(b) = aa
93639⟨a, b | aa=1, abab=aab⟩φ(a) = b, φ(b) = aa
93642⟨a, b | aa=1, abab=baa⟩φ(a) = b, φ(b) = aa
109663⟨a, b | aa=1, aaaaabb=b⟩φ(a) = b, φ(b) = a
109675⟨a, b | aa=1, aaabaab=b⟩φ(a) = b, φ(b) = a
109681⟨a, b | aa=1, aaabbaa=b⟩φ(a) = b, φ(b) = a
109717⟨a, b | aa=1, abaaaab=b⟩φ(a) = b, φ(b) = a
109934⟨a, b | aa=1, aaaaab=bb⟩φ(a) = b, φ(b) = a
109940⟨a, b | aa=1, aaaabb=ab⟩φ(a) = b, φ(b) = a
109946⟨a, b | aa=1, aaabaa=bb⟩φ(a) = b, φ(b) = a
109953⟨a, b | aa=1, aaabba=ba⟩φ(a) = b, φ(b) = a
109960⟨a, b | aa=1, aabaab=ab⟩φ(a) = b, φ(b) = a
109972⟨a, b | aa=1, aabbaa=ab⟩φ(a) = b, φ(b) = a
1010029⟨a, b | aa=1, baaaab=ab⟩φ(a) = b, φ(b) = a
1010209⟨a, b | aa=1, aaaab=abb⟩φ(a) = b, φ(b) = a
1010220⟨a, b | aa=1, aaaba=bba⟩φ(a) = b, φ(b) = a
1010223⟨a, b | aa=1, aaabb=aab⟩φ(a) = b, φ(b) = a
1010226⟨a, b | aa=1, aaabb=baa⟩φ(a) = b, φ(b) = a
1010233⟨a, b | aa=1, aabaa=abb⟩φ(a) = b, φ(b) = a
1010246⟨a, b | aa=1, aabba=aba⟩φ(a) = b, φ(b) = a
1010261⟨a, b | aa=1, abaab=aab⟩φ(a) = b, φ(b) = a
1010264⟨a, b | aa=1, abaab=baa⟩φ(a) = b, φ(b) = a
1010476⟨a, b | aa=1, aabb=aaab⟩φ(a) = b, φ(b) = a
1010480⟨a, b | aa=1, abaa=aabb⟩φ(a) = b, φ(b) = a
1010488⟨a, b | aa=1, abba=aaba⟩φ(a) = b, φ(b) = a
1010503⟨a, b | aa=1, baab=aaab⟩φ(a) = b, φ(b) = a
1126451⟨a, b | aa=1, aaaaabab=b⟩φ(a) = b, φ(b) = aa
1126473⟨a, b | aa=1, aaabaaab=b⟩φ(a) = b, φ(b) = aa
1126479⟨a, b | aa=1, aaababaa=b⟩φ(a) = b, φ(b) = aa
1126557⟨a, b | aa=1, abaaaaab=b⟩φ(a) = b, φ(b) = aa
1126988⟨a, b | aa=1, aaaabab=ab⟩φ(a) = b, φ(b) = aa
1127008⟨a, b | aa=1, aaababa=ba⟩φ(a) = b, φ(b) = aa
1127031⟨a, b | aa=1, aabaaab=ab⟩φ(a) = b, φ(b) = aa
1127043⟨a, b | aa=1, aababaa=ab⟩φ(a) = b, φ(b) = aa
1127169⟨a, b | aa=1, baaaaab=ab⟩φ(a) = b, φ(b) = aa
1127507⟨a, b | aa=1, aaaaab=bab⟩φ(a) = b, φ(b) = aa
1127531⟨a, b | aa=1, aaabaa=bab⟩φ(a) = b, φ(b) = aa
1127535⟨a, b | aa=1, aaabab=aab⟩φ(a) = b, φ(b) = aa
1127538⟨a, b | aa=1, aaabab=baa⟩φ(a) = b, φ(b) = aa
1127568⟨a, b | aa=1, aababa=aba⟩φ(a) = b, φ(b) = aa
1127613⟨a, b | aa=1, abaaab=aab⟩φ(a) = b, φ(b) = aa
1127616⟨a, b | aa=1, abaaab=baa⟩φ(a) = b, φ(b) = aa
1128039⟨a, b | aa=1, aaaab=abab⟩φ(a) = b, φ(b) = aa
1128060⟨a, b | aa=1, aaaba=baba⟩φ(a) = b, φ(b) = aa
1128086⟨a, b | aa=1, aabaa=abab⟩φ(a) = b, φ(b) = aa
1128093⟨a, b | aa=1, aabab=aaab⟩φ(a) = b, φ(b) = aa
1128096⟨a, b | aa=1, aabab=abaa⟩φ(a) = b, φ(b) = aa
1128158⟨a, b | aa=1, ababa=aaba⟩φ(a) = b, φ(b) = aa
1128225⟨a, b | aa=1, baaab=aaab⟩φ(a) = b, φ(b) = aa