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4. The goal of this exercise is to prove the converse to Mashke's theorem in the special case of a p-group. (a) Let p be a prime, F a field of characteristic p, and G a group of order pk for some k≥ 0. Prove that the only irreducible representation of G is the trivial representation. (Hint: Show it first for k= 1, then proceed by induction in k. It may be useful to know that if G is non-trivial, then Z(G) is non-trivial and contains an element of order p.) (b) Let p be a prime, G a group of order pk for k> 0 and F a field of characteristic p. Prove that the regular representation is not completely reducible.

4. The goal of this exercise is to prove the converse to Mashke's theorem in the special case of a p-group. (a) Let p be a prime, F a field of characteristic p, and G a group of order pk for some k≥ 0. Prove that the only irreducible representation of G is the trivial representation. (Hint: Show it first for k= 1, then proceed by induction in k. It may be useful to know that if G is non-trivial, then Z(G) is non-trivial and contains an element of order p.) (b) Let p be a prime, G a group of order pk for k> 0 and F a field of characteristic p. Prove that the regular representation is not completely reducible.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. The goal of this exercise is to prove the converse to Mashke's theorem in the special case of a p-group. (а) Let some k > 0. Prove that the only irreducible representation of G is the trivial representation. (Hint: Show it first for k = 1, then proceed by induction in k. It may be useful to know that if G is non-trivial, then Z(G) is non-trivial and be a prime, F a field of characteristic p, and G a group of order p for contains an element of order p.) (b) Let p be a prime, G a group of order p for k > 0 and F a field of characteristic p. Prove that the regular representation is not completely reducible.
Transcribed Image Text:4. The goal of this exercise is to prove the converse to Mashke's theorem in the special case of a p-group. (а) Let some k > 0. Prove that the only irreducible representation of G is the trivial representation. (Hint: Show it first for k = 1, then proceed by induction in k. It may be useful to know that if G is non-trivial, then Z(G) is non-trivial and be a prime, F a field of characteristic p, and G a group of order p for contains an element of order p.) (b) Let p be a prime, G a group of order p for k > 0 and F a field of characteristic p. Prove that the regular representation is not completely reducible.
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