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I am seeking to gain a good understanding of group presentations
Currently I have the following general question:
"Does a group presentation completely determine a particular group?"
The textbooks I am reading seem to indicate that a group presentation does actually determine/specify the group.
For example on page 31 of James and Liebeck: Representations and Characters of Groups we find:
"Let G be the dihedral group [itex] D_{2n} = D_8 = <a,b: a^4 = b^2 = 1, b^{1}ab = a^{1}> [/itex]
Is this a complete specification of the dihedral group  i.e. does this presentation completely determine or specify the dihedral group [itex] D_8 [/itex]?
Surely it does not  because we additionally need to know that (or do we?)
a = (1 2 3 4) [ rotation of a sqare clockwise through the origin  see attached]
and
b = (2 4) [reflection about the line of symmetry through vertex 1 and the origin  see attached]
Possibly we also need to know that the elements of the group are
[itex] D_8 = \{ 1, a, a^2, a^3, b, ba, ba^2, ba^3 \} [/itex]
but I suspect this can be deduced from the given relations.
Currently I have the following general question:
"Does a group presentation completely determine a particular group?"
The textbooks I am reading seem to indicate that a group presentation does actually determine/specify the group.
For example on page 31 of James and Liebeck: Representations and Characters of Groups we find:
"Let G be the dihedral group [itex] D_{2n} = D_8 = <a,b: a^4 = b^2 = 1, b^{1}ab = a^{1}> [/itex]
Is this a complete specification of the dihedral group  i.e. does this presentation completely determine or specify the dihedral group [itex] D_8 [/itex]?
Surely it does not  because we additionally need to know that (or do we?)
a = (1 2 3 4) [ rotation of a sqare clockwise through the origin  see attached]
and
b = (2 4) [reflection about the line of symmetry through vertex 1 and the origin  see attached]
Possibly we also need to know that the elements of the group are
[itex] D_8 = \{ 1, a, a^2, a^3, b, ba, ba^2, ba^3 \} [/itex]
but I suspect this can be deduced from the given relations.
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