MHB What Is the Upper Bound of Groups of Order in Finite Group Theory?

pauloromero1983
Messages
2
Reaction score
0
In the context of group theory, there's a theorem that states that for a given positive integer \(n\) there exist finitely different types of groups of order \(n\). Notice that the theorem doesn´t say anything of how many groups there are, only states that such groups exist. In the proof of this statement, they define a map \(f:G\times G \rightarrow X\) where \(X\) is a set with \(n\) elements. Defining a group structure in the same map by means of the product rule \(f(g_{1})f(g_{2})=f(g_{1}g_{2})\), where \(g_{1}, g_{2}\) belong to \(G\) they arrive to the following conclusion: there's an upper bound on the number of different groups of order \(n\), namely: \(n^{n^{2}}\)

My question is how to arrive to such conclusion. I am aware that, for every ordered pair of \(G\times G\) there's \(n\) images (since \(X\) was assumed to have \(n\) elements). For a concrete example, let be \(G\) a group of 2 elements. Then, there are 4 ordered pairs. Each pair has 2 images, so the total number of maps would be 4*2=8. However, by use of the relation \(n^{n^{2}}\) we get \(2^{2^{2}}=16\), i.e, there are 16 different maps, not 8. I am missing something here, but I don't know what exactly what the error is.
 
Physics news on Phys.org
Hi pauloromero1983,

In the example you gave for each element of $G\times G$ there are 2 choices in $X$ to which the element can be mapped. Since the total number of elements in $G\times G$ is 4, the total number of possible mappings is $2\times 2\times 2\times 2 = 2^{4} = 16.$

In general, if $A$ and $B$ are finite sets, then there are $|B|^{|A|}$ different mappings/functions from $A$ to $B$. Does this help answer your question?
 
ok, I think I understand now, thank you.
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...

Similar threads

Back
Top