Zappa–Szép product

Apr 21, 2026 - 08:44
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Zappa–Szép product

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← Previous revision Revision as of 03:12, 21 April 2026
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* α(''e'', ''h'') = ''h'' and β(''k'', ''e'') = ''k'' for all ''h'' in ''H'' and ''k'' in ''K''.
* α(''e'', ''h'') = ''h'' and β(''k'', ''e'') = ''k'' for all ''h'' in ''H'' and ''k'' in ''K''.
* α(''k''1''k''2, ''h'') = α(''k''1, α(''k''2, ''h''))
* α(''k''1''k''2, ''h'') = α(''k''1, α(''k''2, ''h''))
* β(''k'', ''h''1''h''2) = β(β(''k'', ''h''1), ''h''2)
* β(''k'', ''h''1''h''2) = β(β(''k'', ''h''1), ''h''2)
* α(''k'', ''h''1''h''2) = α(''k'', ''h''1) α(β(''k'', ''h''1), ''h''2)
* α(''k'', ''h''1''h''2) = α(''k'', ''h''1)α(β(''k'', ''h''1), ''h''2)
* β(''k''1''k''2, ''h'') = β(''k''1, α(''k''2, ''h'')) β(''k''2, ''h'')
* β(''k''1''k''2, ''h'') = β(''k''1, α(''k''2, ''h''))β(''k''2, ''h'')


for all ''h''1, ''h''2 in ''H'', ''k''1, ''k''2 in ''K''. From these, it follows that
for all ''h''1, ''h''2 in ''H'', ''k''1, ''k''2 in ''K''. From these, it follows that
* For each ''k'' in ''K'', the mapping ''h'' {{mapsto}} α(''k'', ''h'') is a [[bijection]] of ''H''.
* For each ''k'' in ''K'', the mapping ''h'' {{mapsto}} α(''k'', ''h'') is a [[bijection]] of ''H''.
* For each ''h'' in ''H'', the mapping ''k'' {{mapsto}} β(''k'', ''h'') is a bijection of ''K''.
* For each ''h'' in ''H'', the mapping ''k'' {{mapsto}} β(''k'', ''h'') is a bijection of ''K''.
(Indeed, suppose α(''k'', ''h''1) = α(''k'', ''h''2). Then ''h''1 = α(''k''−1''k'', ''h''1) = α(''k''−1, α(''k'', ''h''1)) = α(''k''−1, α(''k'', ''h''2)) = ''h''2. This establishes injectivity, and for surjectivity, use ''h'' = α(''k'', α(''k''−1, ''h'')).)
(Indeed, suppose α(''k'', ''h''1) = α(''k'', ''h''2). Then ''h''1 = α(''k''−1''k'', ''h''1) = α(''k''−1, α(''k'', ''h''1)) = α(''k''−1, α(''k'', ''h''2)) = ''h''2. This establishes injectivity, and for surjectivity, use ''h'' = α(''k'', α(''k''−1,''h'')).)


More concisely, the first three properties above assert the mapping α : ''K'' × ''H'' → ''H'' is a [[Group action (mathematics)#Definition|left action]] of ''K'' on (the underlying set of) ''H'' and that β : ''K'' × ''H'' → ''K'' is a [[Group action (mathematics)#Definition|right action]] of ''H'' on (the underlying set of) ''K''. If we denote the left action by ''h'' → ''k''''h'' and the right action by ''k'' → ''k''''h'', then the last two properties amount to ''k''(''h''1''h''2) = ''k''''h''1 ''k''''h''1''h''2 and (''k''1''k''2)''h'' = ''k''1''k''2''h'' ''k''2''h''.
More concisely, the first three properties above assert the mapping α : ''K'' × ''H'' → ''H'' is a [[Group action (mathematics)#Definition|left action]] of ''K'' on (the underlying set of) ''H'' and that β : ''K'' × ''H'' → ''K'' is a [[Group action (mathematics)#Definition|right action]] of ''H'' on (the underlying set of) ''K''. If we denote the left action by ''h'' → ''k''''h'' and the right action by ''k'' → ''k''''h'', then the last two properties amount to ''k''(''h''1''h''2) = ''k''''h''1 ''k''''h''1''h''2 and (''k''1''k''2)''h'' = ''k''1''k''2''h'' ''k''2''h''.
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Turning this around, suppose ''H'' and ''K'' are groups (and let ''e'' denote each group's identity element) and suppose there exist mappings α : ''K'' × ''H'' → ''H'' and β : ''K'' × ''H'' → ''K'' satisfying the properties above. On the [[cartesian product]] ''H'' × ''K'', define a multiplication and an inversion mapping by, respectively,
Turning this around, suppose ''H'' and ''K'' are groups (and let ''e'' denote each group's identity element) and suppose there exist mappings α : ''K'' × ''H'' → ''H'' and β : ''K'' × ''H'' → ''K'' satisfying the properties above. On the [[cartesian product]] ''H'' × ''K'', define a multiplication and an inversion mapping by, respectively,


* (''h''1, ''k''1) (''h''2, ''k''2) = (''h''1 α(''k''1, ''h''2), β(''k''1, ''h''2) ''k''2)
* (''h''1, ''k''1)(''h''2, ''k''2) = (''h''1α(''k''1, ''h''2), β(''k''1, ''h''2)''k''2)
* (''h'', ''k'')−1 = (α(''k''−1, ''h''−1), β(''k''−1, ''h''−1))
* (''h'', ''k'')−1 = (α(''k''−1,''h''−1), β(''k''−1,''h''−1))


Then ''H'' × ''K'' is a group called the external '''Zappa–Szép product''' of the groups ''H'' and ''K''. The [[subset]]s ''H'' × {''e''} and {''e''} × ''K'' are subgroups [[group isomorphism|isomorphic]] to ''H'' and ''K'', respectively, and ''H'' × ''K'' is, in fact, an internal Zappa–Szép product of ''H'' × {''e''} and {''e''} × ''K''.
Then ''H'' × ''K'' is a group called the external '''Zappa–Szép product''' of the groups ''H'' and ''K''. The [[subset]]s ''H'' × {''e''} and {''e''} × ''K'' are subgroups [[group isomorphism|isomorphic]] to ''H'' and ''K'', respectively, and ''H'' × ''K'' is, in fact, an internal Zappa–Szép product of ''H'' × {''e''} and {''e''} × ''K''.
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