group algebras of D4 and Q8.. please help!!

[algebrist]

ok this is my problem:
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for
D4 - the dihedral group of order 8
and
Q8 - quaternion group of order 8

describe the group algebra kG (for a big enough k so that Masche thm. holds), both its algebra and coalgebra structure.
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please if you have any suggestions how should i procede i'll be glad to see them

[matt grime]

First you must tell us over what field you want the algebra defined. If the field has characteristic not equal to 2 the answer is almost trivial by wedderburn's structure theorem (it is FxFxFxFx(M_2(F))x(M_2(F)) where F is the field I think, by complete reducibility of the group's representation theory). Over char 2 it's a little harder.

[matt grime]

Sorry, I omitted to say, that the *size* of k is unimportant to describe the algebra structure, merely its characteristic (if it is a field, a ring is different again). It is a full matrix algebra so its structure and coalgebra structure are easy to describe (if the char is not 2).