Chapter Questions
Prove that there is no simple group of order $210=2 \cdot 3 \cdot 5 \cdot 7$.
Prove that there is no simple group of order $280=2^{3} \cdot 5 \cdot 7$.
Prove that there is no simple group of order $216=2^{3} \cdot 3^{3}$.
Prove that there is no simple group of order $300=2^{2} \cdot 3 \cdot 5^{2}$.
Prove that there is no simple group of order $525=3 \cdot 5^{2} \cdot 7$.
Prove that there is no simple group of order $540=2^{2} \cdot 3^{3} \cdot 5$.
Prove that there is no simple group of order $528=2^{4} \cdot 3 \cdot 11$.
Prove that there is no simple group of order $315=3^{2} \cdot 5 \cdot 7$.
Prove that there is no simple group of order $396=2^{2} \cdot 3^{2} \cdot 11$.
Prove that there is no simple group of order $n$, where $201 \leq$ $n \leq 235$ and $n$ is not prime.
Without using the Generalized Cayley Theorem or its corollaries, prove that there is no simple group of order 112 .
Without using the $2 \cdot$ Odd Test, prove that there is no simple group of order 210 .
You may have noticed that all the "hard integers" are even. Choose three odd integers between 200 and 1000 . Show that none of these is the order of a simple group unless it is prime.
Show that there is no simple group of order $p q r$, where $p, q$, and $r$ are primes $(p, q$, and $r$ need not be distinct).
Show that $A_{5}$ does not contain a subgroup of order 30,20 , or 15 .
Prove that that $A_{6}$ has no subgroup of order 120 .
Prove that there is no simple group of order $120=2^{3} \cdot 3 \cdot 5$. (This exercise is referred to in this chapter.)
Prove that if $G$ is a finite group and $H$ is a proper normal subgroup of largest order, then $G / H$ is simple.
Suppose that $H$ is a subgroup of a finite group $G$ and that $|H|$ and $(|G: H|-1) !$ are relatively prime. Prove that $H$ is normal in $G$. What does this tell you about a subgroup of index 2 in a finite group?
Suppose that $p$ is the smallest prime that divides $|G|$. Show that any subgroup of index $p$ in $G$ is normal in $G$.
Prove that the only nontrivial proper normal subgroup of $S_{5}$ is $A_{5}$. (This exercise is referred to in Chapter 32.)
Prove that a simple group of order 60 has a subgroup of order 6 and a subgroup of order 10 .
Show that $P S L\left(2, Z_{7}\right)=S L\left(2, Z_{7}\right) / Z\left(S L\left(2, Z_{7}\right)\right)$, which has order 168 , is a simple group. (This exercise is referred to in this chapter.)
Show that the permutations (12) and (12345) generate $S_{5}$.
Suppose that a subgroup $H$ of $S_{5}$ contains a 5 -cycle and a 2 -cycle. Show that $H=S_{5} .$ (This exercise is referred to in Chapter $32 .$ )
Suppose that $G$ is a finite simple group and contains subgroups $H$ and $K$ such that $|G: H|$ and $|G: K|$ are prime. Show that $|H|=|K|$.
Show that (up to isomorphism) $A_{5}$ is the only simple group of order 60. (This exercise is referred to in this chapter.)
Prove that a simple group cannot have a subgroup of index 4 .
Prove that there is no simple group of order $p^{2} q$, where $p$ and $q$ are odd primes and $q>p$
If a simple group $G$ has a subgroup $K$ that is a normal subgroup of two distinct maximal subgroups, prove that $K=\{e\}$.
Show that a finite group of even order that has a cyclic Sylow 2 subgroup is not simple.
Show that $S_{5}$ does not contain a subgroup of order 40 or 30 .