MHB How many homomorphisms are there from $\mathbb{Z}_4$ to $S_4$?

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Homomorphisms
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :i

How many homomorphism $f:\mathbb{Z}_4\rightarrow S_4$ are there?

Do we have to find how many permutations of $S_4$ have order that divides $4$ ?

We have 1 identity (order 1), 6 transpositions (order 2), 3 products of two disjoint transpositions (order 2), 6 4-cycles (order 4).

So in total we have 1 + 6 + 3 + 6 = 16 elements of $S_4$ that have order that divides 4, right?

Does this mean that there are $16$ homomorphisms? (Wondering)
 
Physics news on Phys.org
mathmari said:
Hey! :i

How many homomorphism $f:\mathbb{Z}_4\rightarrow S_4$ are there?

Do we have to find how many permutations of $S_4$ have order that divides $4$ ?

We have 1 identity (order 1), 6 transpositions (order 2), 3 products of two disjoint transpositions (order 2), 6 4-cycles (order 4).

So in total we have 1 + 6 + 3 + 6 = 16 elements of $S_4$ that have order that divides 4, right?

Does this mean that there are $16$ homomorphisms? (Wondering)

Yep. (Nod)

For a homomorphism we need that $f(a)f(b)=ab$ for any $a,b$.
The structure of $\mathbb Z_4$ is fully identified by its generator $1$.
That means that it suffices indeed to look at $f(1)$, which fully identifies the homomorphism, and in particular verify that $f(1)^4=\text{id}$. (Nerd)
 
I like Serena said:
Yep. (Nod)

For a homomorphism we need that $f(a)f(b)=ab$ for any $a,b$.
The structure of $\mathbb Z_4$ is fully identified by its generator $1$.
That means that it suffices indeed to look at $f(1)$, which fully identifies the homomorphism, and in particular verify that $f(1)^4=\text{id}$. (Nerd)

Ok! Thank you! (Yes)
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'
This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
Back
Top