Notes - Groups HT23, Homomorphisms

Created: February 17, 2023

Flashcards

What is an isomorphism $\phi$ between two groups $G$ and $H$?

A bijection such that

\[\phi(g_1 \ast g_2) = \phi(g_1) \ast \phi(g_2)\]

What does it mean for two groups to be isomorphic?

There is an isomorphism between them.

What is true about $G$ if $G$ has prime order?
\[G\cong C_n\]

What is true about $G$ if $ \vert G \vert = 2p$ where $p$ prime?
\[G \cong C_{2p} \text{ or } D_{2p}\]

What is true about the elements of $G$ if $G$ has even order?

It contains an element of order $2$.

When proving that if $G$ has even order, then it contains an element of order $2$, what equivalence relation do you define to partition the group into collections of one or two elements?
\[x \sim y \iff x = y \text{ or } y = x^{-1}\]

Let $G$ and $H$ be two groups. What is a homomorphism?

A map $f : G \to H$ such that $f(g _ 1 \ast g _ 2) = f(g _ 1) f(g _ 2)$ for all $g _ 1, g _ 2 \in G$.

What is a homomorphism from $G$ to itself called?

An automorphism.

What is the automorphism group $\text{Aut}(G)$ of a group $G$?

The set of all automorphisms on $G$ under function composition.

Suppose $f : G \to H$ is a group homomorphism and $g \in G$ has order $n$. What can you deduce?
\[o(f(g)) | n\]

What is the trivial homomorphism on a group?

The map that sends every element to the identity.

Can you give the automorphism on a group $G$ that represents conjugation by $g$?
\[\theta_g(h) = g^{-1} h g\]

Let $f : G \to H$ be a group homomorphism. Can you define $\text{Im} f$?
\[\\{f(g) : g \in G\\} \subseteq H\]

Let $f : G \to H$ be a group homomorphism. Can you define $\text{ker} f$?
\[\\{g \in G : f(g) = e_H\\} \subseteq G\]

What does it mean for a homomorphism $\phi : G \to H$ to be constant on a coset of $\ker \phi$ and to take different values on different cosets?

For all $g _ 1, g _ 2 \in G$, $\phi(g _ 1) = \phi(g _ 2)$ if and only if $g _ 1 \ker \phi = g _ 2 \ker \phi$.

Can you quickly prove that a homomorphism $\phi : G \to H$ will be constant on cosets of $\ker \phi$ and takes different values on different cosets?
\[\begin{aligned} \phi(g_1) = \phi(g_2) &\iff \phi(g_2^{-1} g_1) = e_H \\\\ &\iff g_2^{-1}g_1 \in \ker \phi \\\\ &\iff g_1 \ker \phi = g_2 \ker \phi \end{aligned}\]

There’s a result in linear algebra that a linear transformation $T$ is injective if and only if $\text{nullity}(T) = 0$. What’s the analoguous result about homomorphisms?

$f : G \to H$ is an isomorphism if and only if $\ker f = \{ e _ H \}$.

What fact connecting the kernel of a homomorphism $\phi : G \to H$ and cosets allows you to prove that $\phi$ is an isomorphism if and only if its kernel is $\{e _ H\}$?

For all $g _ 1, g _ 2 \in G$, $\phi(g _ 1) = \phi(g _ 2)$ if and only if $g _ 1 \ker \phi = g _ 2 \ker \phi$.

Quickly justify that $\ker \phi \trianglelefteq G$?
\[\phi(g^{-1} h g) = \phi(g)^{-1}\phi(g) = e_H \implies g^{-1}g h \in \ker \phi\]

Proofs

Prove that an isomorphism $\phi : G \to H$ between groups $(G, \ast _ G)$ and $(H, \ast _ H)$ has

\[\begin{aligned} &\phi(e_g) = e_h \\\\ &\phi(g^{-1}) = \phi(g)^{-1} \end{aligned}\]

Todo.

Prove that any group of prime order $p$ is isomorphic to $C _ p$.

Todo.

Prove that any group of even order has an element of order $2$.

Todo.

Prove that if $f : G \to H$ is a group homomorphism and $g \in G$ has order $n$, then

\[o(f(g)) | n\]

Todo (groups and group actions, page 50)

Prove that if $G$ is a group and $g, h \in G$ are conjugate, then $o(g) = o(h)$ and $g^{-1}$ and $h^{-1}$ are conjugate.

Todo (groups and group actions, page 51). Consider $o(g) \vert o(h)$ and $o(h) \vert o(g)$.



Related posts