Dear readers,

Below are some links that may provide motivation to study “Abstract Algebra”.

.

Why do we need to study abstract algebra?

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# Category: Group Theory

## Epsilon-motivation for studying “Abstract Algebra”

## How to check whether a polynomial has roots in the group of integers modulo n?

## Prove that each term in the lower central series of a group G is characteristic in G.

## Prove that each term in the upper central series of a group G is characteristic in G.

## Characteristic subgroups and its properties

## Computation of “CENTER OF A DIHEDRAL GROUP”

## Proof: group of order 4 is abelian

Its all about pure mathematics.

Dear readers,

Below are some links that may provide motivation to study “Abstract Algebra”.

.

Why do we need to study abstract algebra?

Dear readers, I am writing this post on a question put by my student. Download the solution for details. Solution

To see the proof

To see the proof

A subgroup $latex H$ of a group $latex G$ is called characteristic in $latex G$, denoted $latex H$ char $latex G$, if every automorphism of $latex G$ maps $latex H$ to itself, i.e., $latex \sigma (H) = H$ for all $latex \sigma \in Aut(G)$.

Results concerning characteristic subgroups which we shall use later (and whose proofs are relegated to the exercises) are

(1) characteristic subgroups are normal,

(2) if $latex H$ is the unique subgroup of $latex G$ of a given order, then $latex H$ is characteristic in $latex G$, and

(3) if $latex K$ char $latex H$ and $latex H \trianglelefteq G$, then $latex K$ $latex \trianglelefteq G$ (so although “normality” is not a transitive property (i.e., a normal subgroup of a normal subgroup need not be normal), a: characteristic subgroup of a normal subgroup is normal).

(4) the relation of “being characteristic” is transitive.

For proofs click

In the link below, the center of Dihedral group D_n for any positive integer n is computed. It is proved that the

Center of $latex D_n= {1}$, if n is odd &

Center of $latex D_n =\{1, r^{n/2} \}$, if n is even

For proof