Top and bottom composition factors of $M$ are isomorphic

Let $$k$$ be a field and $$N$$ a finite group. Let $$M$$ be a projective indecomposable $$kN$$-module. Since the algebra $$kN$$ is symmetric, it follows that the top and bottom composition factors of $$M$$ are isomorphic. In particular, there is a nonzero endomorphism of
$$M$$ sending $$M$$ onto the socle $$operatorname{soc}(M)$$.

I cannot see the connection here. How does being symmetric implies composition factors? Any help would be appreciated!

MathOverflow Asked by user666 on January 2, 2021

For every Frobenius algebra $$A$$ there is a bijection $$pi$$ such that $$top(P_i) cong soc(P_{pi (i)})$$ when $$P_i$$ denote the indecomposable projective $$A$$-modules. Being symmetric implies that $$A$$ is weakly symmetric (meaning that $$pi$$ is the identity). Thus top and socle of every $$P_i$$ coincide which is what you asked for when I understand your question correct. For proofs and more on this see the book "Frobenius algebras I" in chapter IV. by Skowronski and Yamagata. When M is an indecomposable projective $$A$$-module, let $$S:=soc(M)$$ be the socle of $$M$$. Then we have a surjective map $$M rightarrow S$$ (since top and socle of $$M$$ coincide) that induces an isomorphism $$top(M)=M/rad(M) rightarrow S$$. Thus we have a surjective map $$M rightarrow soc(M)$$.

Correct answer by Mare on January 2, 2021

Related Questions

Are universal geometric equivalences of DM stacks affine?

0  Asked on January 17, 2021 by harry-gindi

Barycentric coordinates of weighted edges

0  Asked on January 17, 2021 by manfred-weis

Question on limit in probability of the ratio of max to min of 2 sequences of non-ive, continuous iid random variables with support $[0, infty).$

1  Asked on January 16, 2021 by learning-math

PhD dissertations that solve an established open problem

18  Asked on January 16, 2021

Commutant of the conjugations by unitary matrices

3  Asked on January 16, 2021 by jochen-glueck

Constructing intertwiners between representations of compact quantum groups

1  Asked on January 15, 2021

A certain property for Heegaard splittings

1  Asked on January 15, 2021 by no_idea

When have we lost a body of mathematics because errors were found?

10  Asked on January 14, 2021 by edmund-harriss

Prove that there are no composite integers $n=am+1$ such that $m | phi(n)$

1  Asked on January 13, 2021 by david-jones

Kernel of the map $mathbb{C}[G]^U to mathbb{C}[U^+]$

0  Asked on January 13, 2021 by jianrong-li

Finite fast tests for periodicity of certain matrices

1  Asked on January 12, 2021

Are these two kernels isomorphic groups?

0  Asked on January 12, 2021 by francesco-polizzi

Definition of subcoalgebra over a commutative ring

2  Asked on January 11, 2021 by user839372

Riesz Representation Theorem for $L^2(mathbb{R}) oplus L^2(mathbb{T})$?

0  Asked on January 10, 2021 by goulifet

Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

0  Asked on January 10, 2021 by stabilo

Cut points and critical points of the exponential map

0  Asked on January 9, 2021 by longyearbyen

Eigendecomposition of $A=I+BDB^H$

0  Asked on January 9, 2021 by user164237