What is the structure of group algebras for D4 and Q8?

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In summary, the group algebras of D4 and Q8, denoted as kG, are being discussed. The group algebra kG is defined over a field, with different structures depending on the characteristic of the field. If the field has characteristic not equal to 2, the group algebra is FxFxFxFx(M_2(F))x(M_2(F)), where F is the field. If the characteristic is 2, the group algebra is a full matrix algebra. The algebra and coalgebra structure of kG are straightforward to describe in both cases.
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algebrist
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group algebras of D4 and Q8.. please help!

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
 
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  • #2
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.
 
  • #3
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).
 

What is a group algebra?

A group algebra is a mathematical structure that combines the concepts of a group and an algebra. It is a vector space over a field, with an additional multiplication operation defined by the group operation.

What is D4?

D4 is a group, also known as the dihedral group of order 8. It consists of the symmetries of a square, including rotations and reflections.

What is Q8?

Q8 is also a group, known as the quaternion group of order 8. It is a non-abelian group, meaning that the order of multiplication matters.

What is the group algebra of D4?

The group algebra of D4 is a mathematical structure that combines the elements of D4 with the elements of a chosen field, such as the real numbers. It is denoted as D4[F], where F represents the chosen field.

What is the group algebra of Q8?

The group algebra of Q8 is a mathematical structure that combines the elements of Q8 with the elements of a chosen field, such as the real numbers. It is denoted as Q8[F], where F represents the chosen field.

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