Group Theory: Axioms, Symmetries, and Structural Ideas

All flashcards

What is a group?

A set with a closed, associative binary operation, an identity element, and an inverse for every element.

What is a group homomorphism?

A function between groups that preserves their operations.

What is an isomorphism of groups?

A bijective homomorphism; it identifies groups with the same structure.

What is the order of an element?

The least positive integer $n$ such that $a^n=e$, if such an integer exists.

What makes a subgroup normal?

Its left and right cosets agree for every element of the group.

Quiz questions

1. How many elements are in the group of symmetries of a square under composition?Answer: 8
2. For groups with operations $\cdot$ and $*$, which condition defines a homomorphism $\varphi:G\to H$?Answer: $\varphi(a\cdot b)=\varphi(a)*\varphi(b)$
3. What condition on a subgroup $N$ of $G$ ensures that its cosets form the quotient group $G/N$?Answer: $N$ is normal in $G$.
4. If $G$ is finite, what does Lagrange's theorem say about the order of a subgroup $H$?Answer: The order of $H$ divides the order of $G$.
5. For prime $p$, why do the nonzero residue classes modulo $p$ form a group under multiplication?Answer: Every nonzero class has a multiplicative inverse modulo $p$.