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(2) Let G be a cyclic group of order n. Show that, up to isomorphism, G admits exactly n one-dimensional complex representations.

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(1)

Algebra III/Introduction to Algebra III: Representation Theory Due: Please upload solutions to NUCT by Tuesday, April 28, 2020.

Problem 1. (1) Let G be a group and let π : G → C

×

be a one-dimensional complex representation of a group G. Show that if an element g ∈ G has (finite) order n, then π(g) ∈ C

×

is an nth root of unity.

(2) Let G be a cyclic group of order n. Show that, up to isomorphism, G admits exactly n one-dimensional complex representations.

[Hint: First construct n one-dimensional complex representations π

i

: G → C

×

, 0 ≤ i < n, such that π

i

' π

j

implies that i = j. Next show that if π : G → C

×

is any one-dimensional complex representation, then π ' π

i

for some 0 ≤ i < n.]

Problem 2. Find, up to isomorphism, all one-dimensional complex representations of the infinite cyclic group G = ( Z , +).

1

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