Michigan State | Problems in Mathematics

Group Theory

06/27/2017

Every Finite Group Having More than Two Elements Has a Nontrivial Automorphism

Problem 495

Prove that every finite group having more than two elements has a nontrivial automorphism.

(Michigan State University, Abstract Algebra Qualifying Exam)

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Group Theory

06/22/2017

A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal

Problem 470

Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.

(Michigan State University, Abstract Algebra Qualifying Exam)

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