Intereting Posts

natural isomorphism in linear algebra
Basic Subgroup Conditions
Why is the second derivative of an inflection point zero?
If $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid$ then vectors $a_1,\dots ,a_n \in \mathbb{R}^n$ are linearly indendent.
Liouville function and perfect square
question about construction of real numbers
Let $4$ and $5$ be the only eigenvalues of $T$. Show $T^2-9T + 20I = 0$ , T is self adjoint.
Symmetry of Grassmanians
Integrable – martingale
Seeking a more direct proof for: $m+n\mid f(m)+f(n)\implies m-n\mid f(m)-f(n)$
Measures: Atom Definitions
Why does spectral norm equal the largest singular value?
If a measure is semifinite, then there are sets of arbitrarily large but finite measure
Difference of order statistics in a sample of uniform random variables
$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$

We know that if $G$ be a finite group and $F$ be an algebraically closed field whose characteristic does not divide the order of $G$, then the number of inequivalent irreducible $F$-representations of $G$ equals the class number of $G$.

Now, if we suppose that the field $F$ is not necessarily an algebraically closed field but its characteristic does not divide the order of $G$, is it true that the number of inequivalent irreducible $F$-representations of $G$ cannot exceed the class number of $G$?

If the answer is yes, how can we prove that?

- Why is the Monster group the largest sporadic finite simple group?
- Subgroups whose order is relatively prime to the index of another subgroup
- Examples of finite nonabelian groups.
- Possible Class equation for a group
- Modern Books on Group Theory in German or French
- Finite group with elements of given order

I know that each irreducible representation of $G$ over $F$ is completely reducible by Maschke’s Theorem because the characteristic of $F$ does not divide the order of $G$ and $G$ is finite. Also, each representation of $G$ over $F$ can be considered as a representation of $G$ over $\overline{F}$, where $\overline{F}$ is an algebraic closure of $F$. But I have no idea about how they can help.

I will be so grateful for any answers and comments.

- Find a finite generating set for $Gl(n,\mathbb{Z})$
- If there are injective homomorphisms between two groups in both directions, are they isomorphic?
- Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$
- An elementary question regarding a multiplicative character over finite fields
- The free group $F_2$ contains $F_k$
- Khan academy for abstract algebra
- How to prove that the normalizer of diagonal matrices in $GL_n$ is the subgroup of generalized permutation matrices?
- Classifying the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$
- Prove that Z(G) which is the center of G is a subgroup of G
- Order of Cyclic Subgroups

Yes, the number of irreducible representations cannot exceed the class number (in the case of caprice [autocorrect error. Should be “coprime”] characteristic): Consider the case of an $F$-irreducible module $M$ that becomes reducible over $\bar F$. Then $M$ has an $\bar F$-submodule $N$, and $M$ is simply the direct sum of Galois-conjugates of $N$ (the Galois-sum is an $F$-invariant submodule). That shows that the number of $F$-irreducible representations never exceeds the number of $\bar F$-irreducible representations.

By Maschke’s theorem and the Artin-Wedderburn theorem, the group algebra $k[G]$ decomposes as a finite product $\prod_i M_{n_i}(D_i)$ of matrix algebras over finite-dimensional division algebras $D_i$ over $k$. Here the product runs over all simple modules of $k[G]$.

The center of this product is $\prod_i Z(D_i)$, and hence the number of simple modules is at most the dimension of the center. The center of $k[G]$ has a natural basis given by sums over the conjugacy classes of $G$, and in particular it always has dimension the number of conjugacy classes of $G$.

- Nonconstant polynomials do not generate maximal ideals in $\mathbb Z$
- Writing a Polar Equation for the Graph of an Implicit Cartesian Equation
- First Course in Linear Algebra book suggestions?
- Convert a linear scale to a logarithmic scale
- Normal subgroup where G has order prime
- What is this probability of a random power of 2 beginning with 1?
- Prove that every complex number with modulus 1 and is not -1, has this property
- $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$
- How do we prove $\cos(\pi/5) – \cos(2\pi/5) = 0.5$ ?.
- Regard naturally as modules.
- Equality in rng with no zero divisors.
- fundamental group of the Klein bottle minus a point
- Number of elements in a finite $\sigma$-algebra
- The first Stiefel-Whitney class is zero if and only if the bundle is orientable
- Is a Cauchy sequence – preserving (continuous) function is (uniformly) continuous?