I am seeking to gain a good understanding of group presentations(adsbygoogle = window.adsbygoogle || []).push({});

Currently I have the following general question:

"Does a group presentation completely determine a particular group?"

The textbooks I am readingto indicate that a group presentation does actually determine/specify the group.seem

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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# Group Presentations - do they determine the group

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