Assume I have simple group G and its cyclic Sylow subgroup H. Are there any nice properties of the induced representation ? 


Any remarks "around the situation" (relaxing "simple" "cyclic" "Sylow") are also welcome.


**Example:** Consider [GL\_3(F\_2)](http://en.wikipedia.org/wiki/PSL(2,7)) which is [remarkbly isomorphic](https://mathoverflow.net/questions/37525/what-is-your-favorite-isomorphism/37542#37542) to PSL\_2(F\_7).


It has cyclic Sylow subgroups of order 7 and 3. And irreps with dimensions 1,3,3,6,7,8.


$Ind\_{Sylow 7} = V\_1 \oplus V\_7\oplus 2V\_8$


$Ind\_{Sylow 3} = V\_1 \oplus V\_{3a}\oplus V\_{3b} \oplus 2V\_6\oplus 3 V\_7 \oplus 2V\_8$


Here induction is from trivial character.


This is result of numerical calcultion with character table (hope I did not make mistake).


Is there some advice which can help me to predict the results in advance at least partitialy ?


I am making similar calculations with other groups, using character is a little painful,
if something can be done more easily - would be very helpful.


PS


**Particular question:**
In particular I need to calculate inductions from non-trivial charcater in the cases above,
any guess how to get the results without calculation ? 



Induction from cyclic / Sylow subgroup are there any nice properties ?
Here is a partial answer. (Your assumption that $G$ is simple is irrelevant for these remarks.) Suppose, as in your example, that the Sylow $p$-subgroup $P$ has order $p$ exactly. Then every irreducible character $\chi$ with degree divisible by $p$ appears as a constituent of $\lambda^G$ with multiplicity $\chi(1)/p$ for every irreducible character $\lambda$ of $P$.


There is a deep theory (due to Richard Brauer) which describes in some detail what the irreducible characters are of a group with a Sylow $p$-subgroup $P$ of prime order. To illustrate the power of the theory, I will indicate how to compute the multiplicity of an irreducible character $\chi$ in $(1\_P)^G$ in the special case where $P = {\text C}\_G(P)$. Write $e = |{\text N}\_G(P):P|$, so $e$ divides $p-1$. Then except possibly for $(p-1)/e$ so-called $exceptional$ characters, each irreducible character of $G$ with degree not divisible by $p$ has degree congruent to $\pm1$ mod $p$. If $\chi(1) = kp + \varepsilon$, where $\varepsilon = \pm1$, then the multiplicity of $\chi$ in $(1\_P)^G$ is exactly $k + \varepsilon$. The exceptional characters have equal degrees, and this common degree is congruent mod $p$ to $\pm e$. (Note that if $e = p-1$, there really is no exception.) If $\chi$ is exceptional with degree $kp + \varepsilon e$, where again $\varepsilon = \pm1$, then $\chi$ appears with mutiplicity $k$ in the induced character. This theory can also be used to find the multiplicity of $\chi$ in $\lambda^G$ for nonprincipal $\lambda \in \text{Irr}(P)$. For nonexceptional $\chi$, the answer is $k$, but the answer depends on the particular character $\lambda$ for the exceptional characters.


In fact, Brauer's theory has been generalized by Dade and others to the case where $P$ is cyclic, but is not of prime order. This is not the place to go into detail, however. One reference for much (but not all) of this theory is the book by G. Navarro from Cambridge University Press.


