Tagged Questions
2
votes
3answers
69 views
Show that A is a field which has $9$ elements
For $A=\left\{\left( {\begin{array}{*{20}{c}}
a&b\\
{ - b}&a
\end{array}} \right) | a,b\in\mathbb{Z/3Z}\right\}$ . Show that A is a field which has $9$ elements . $(A^*, .)$ is a cyclic group ...
5
votes
0answers
51 views
Subgroup Structure of $\mathrm{SL}(2, p^2)$, and Its Irreducible Characters
I am taking a course in representation theory of finite groups,and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and ...
6
votes
2answers
130 views
Exponent of $GL(n,q)$.
Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group.
...
3
votes
1answer
54 views
Abelian groups related to $(\mathbb{Z}/p\mathbb{Z})^*$, with order close to $p$ — analogous to elliptic curve groups, but simpler
I'm looking for a construction of a group $H$ that is "a sister" to $(\mathbb{Z}/p\mathbb{Z})^*$, in the following rough sense:
Each element of $H$ can be represented by one or a few elements of ...
6
votes
1answer
107 views
What does this notation mean: $\displaystyle\lim_{\leftarrow} \,\mathbb{Z}/n\mathbb{Z}$?
Just a small notation question from this Wikipedia page:
The absolute Galois group of a finite field $K$ is isomorphic to the group $$\hat{\mathbb{Z}}=\lim_{\leftarrow} \mathbb{Z}/n\mathbb{Z}.$$
...
2
votes
1answer
63 views
Latin squares of even order with all cells only participating in one subsquare.
For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below:
...
1
vote
1answer
324 views
Irreducible representations of a cyclic group over a field of prime order
Consider $G$ a cyclic group of order $n$ with prime $p\not|n$.
How do I construct all the irreducible representations over $\mathbb F_p$?
How many irreducible representations are there and what are ...
1
vote
1answer
200 views
Formal derivatives over finite fields.
I am slightly confused about what formal derivatives over finite fields mean.
Example 1: Consider $f(x)=x^3-2\in \mathbb{F}_7[x]$.
By checking each element of $\mathbb{F}_7$ we easily see that this ...
5
votes
3answers
378 views
Other ways to deduce Cyclicity of the Units of certain groups?
The group of units of the rings $\text{GF}(p^r)$ and $\mathbb{Z}/p^r\mathbb{Z}$ are both cyclic (except for the exception of prime powers are not cyclic when $p=2$ and $r\ge 3$). This is a strong ...
2
votes
1answer
98 views
Extracting the value of $y$ from $x$ in an elliptic curve over a finite field
Given an elliptic curve $y^2 = x^3 + ax + b$ over a finite field $\mathbf{F}_p$, how can I retrieve the value of $y$ given the value of $x$?
My knowledge in this area is quite limited, so I ...
