to classify the indecomposable representations.

Advice for the novice: some arguments in our paper are only sketched and some notions maybe still unfamiliar for a novice. If that is the case, we advice readers to skip these parts on the first reading as they may became more clear later on and they most likely will not influence the basic understanding of the key ideas. The main point of this article is to provide overview of the remarkable proof of classification of representations of Klein group $V_4$ with appreciation of the works of number of people and to show that this proof open doors to study modern group representation where Auslander-Reiten sequences play increasingly important role. We hope that after reading our article a reader will read more texts in the references and possibly go on to further exciting heights in group representation theory.

Indecomposable Representations of Klein’s four group
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We list all the indecomposable representations of $V_4$ below. Note that these are just the finitely generated modules over the group algebra $$kV_{4} \cong k[a, b]/(a^{2}, b^{2})$$ which cannot be written as a sum of strictly smaller modules (much the same way prime numbers cannot be written as product of 