Epsilon-motivation for studying “Abstract Algebra”

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

Why do we need to study abstract algebra?


Characteristic subgroups and its properties

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