Groups HT23, Homomorphisms

Created: February 17, 2023 Read markdown About these notes View in context Study these flashcards

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@Define an isomorphism $\phi$ between two groups $G$ and $H$.

isomorphism-definition

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?

isomorphic-groups-definition

There is an isomorphism between them.

What is true about $G$ if $G$ has prime order?

prime-order-group-cyclic
\[G\cong C _ p\]

What is true about $G$ if $ \vert G \vert = 2p$ where $p$ prime?

order-2p-group-classification
\[G \cong C _ {2p} \text{ or } D _ {2p}\]

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

even-order-element-of-order-2

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?

even-order-proof-equivalence-relation
\[x \sim y \iff x = y \text{ or } y = x^{-1}\]

Let $G$ and $H$ be two groups. @Define a homomorphism.

homomorphism-definition

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?

endomorphism-definition

An automorphism.

@Define the automorphism group $\mathrm{Aut}(G)$ of a group $G$.

automorphism-group-definition

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?

homomorphism-order-divides
\[o(f(g)) \vert n\]

@Define the trivial homomorphism on a group.

trivial-homomorphism-definition

The map that sends every element to the identity.

Can you give the automorphism on a group $G$ that represents conjugation by $g$?

inner-automorphism-conjugation
\[\theta _ g(h) = g^{-1} h g\]

Let $f : G \to H$ be a group homomorphism. @Define $\mathrm{Im} f$.

image-definition
\[\{f(g) : g \in G\} \subseteq H\]

Let $f : G \to H$ be a group homomorphism. @Define $\ker f$.

kernel-definition
\[\{g \in G : f(g) = e _ H\} \subseteq G\]

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?

homomorphism-constant-on-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?

homomorphism-constant-on-cosets-proof
\[\begin{aligned} \phi(g _ 1) = \phi(g _ 2) &\iff \phi(g _ 2^{-1} g _ 1) = e _ H \\ &\iff g _ 2^{-1}g _ 1 \in \ker \phi \\ &\iff g _ 1 \ker \phi = g _ 2 \ker \phi \end{aligned}\]

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

trivial-kernel-criterion

$f : G \to H$ 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 $\{e _ H\}$?

trivial-kernel-criterion-via-cosets

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

Quickly justify that $\ker \phi \trianglelefteq G$?

kernel-normal-justification
\[\phi(g^{-1} h g) = \phi(g)^{-1}\phi(g) = e _ H \implies g^{-1}g h \in \ker \phi\]

Proofs



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