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The Sylow Theorems: exercises

The Sylow Theorems: exercises — from Judson, Abstract Algebra: Theory and Applications.

Practice (29)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

What are the orders of all Sylow \(p\)-subgroups where \(G\) has order \(18\), \(24\), \(54\), \(72\), and \(80\)?

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Hint:
If \(|G| = 18 = 2 \cdot 3^2\), then the order of a Sylow \(2\)-subgroup is \(2\), and the order of a Sylow \(3\)-subgroup is \(9\).
Find all the Sylow \(3\)-subgroups of \(S_4\) and show that they are all conjugate.

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Hint:
The four Sylow \(3\)-subgroups of \(S_4\) are \(P_1 = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\), \(P_2 = \{ (1), (1 \, 2 \, 4), (1 \, 4 \, 2) \}\), \(P_3 = \{ (1), (1 \, 3 \, 4), (1 \, 4 \, 3) \}\), \(P_4 = \{ (1), (2 \, 3 \, 4), (2 \, 4 \, 3) \}\).
Show that every group of order \(45\) has a normal subgroup of order \(9\).
Let \(H\) be a Sylow \(p\)-subgroup of \(G\). Prove that \(H\) is the only Sylow \(p\)-subgroup of \(G\) contained in \(N(H)\).
Prove that no group of order \(96\) is simple.

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Hint:
Since \(|G| = 96 = 2^5 \cdot 3\), \(G\) has either one or three Sylow \(2\)-subgroups by the Third Sylow Theorem. If there is only one subgroup, we are done. If there are three Sylow \(2\)-subgroups, let \(H\) and \(K\) be two of them. Therefore, \(|H \cap K| \geq 16\); otherwise, \(HK\) would have \((32 \cdot 32)/8 = 128\) elements, which is impossible. Thus, \(H \cap K\) is normal in both \(H\) and \(K\) since it has index \(2\) in both groups.
Prove that no group of order \(160\) is simple.
If \(H\) is a normal subgroup of a finite group \(G\) and \(|H| = p^k\) for some prime \(p\), show that \(H\) is contained in every Sylow \(p\)-subgroup of \(G\).
Let \(G\) be a group of order \(p^2 q^2\), where \(p\) and \(q\) are distinct primes such that \(q \nmid p^2 - 1\) and \(p \nmid q^2 - 1\). Prove that \(G\) must be abelian. Find a pair of primes for which this is true.

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Hint:
Show that \(G\) has a normal Sylow \(p\)-subgroup of order \(p^2\) and a normal Sylow \(q\)-subgroup of order \(q^2\).
Show that a group of order \(33\) has only one Sylow \(3\)-subgroup.
Let \(H\) be a subgroup of a group \(G\). Prove or disprove that the normalizer of \(H\) is normal in \(G\).

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Hint:
False.
Let \(G\) be a finite group whose order is divisible by a prime \(p\). Prove that if there is only one Sylow \(p\)-subgroup in \(G\), it must be a normal subgroup of \(G\).
Let \(G\) be a group of order \(p^r\), \(p\) prime. Prove that \(G\) contains a normal subgroup of order \(p^{r-1}\).
Suppose that \(G\) is a finite group of order \(p^n k\), where \(k \lt p\). Show that \(G\) must contain a proper nontrivial normal subgroup.
Let \(H\) be a subgroup of a finite group \(G\). Prove that \(g N(H) g^{-1} = N(gHg^{-1})\) for any \(g \in G\).
Prove that a group of order \(108\) must have a proper nontrivial normal subgroup.
Classify all the groups of order \(175\) up to isomorphism.
Show that every group of order \(255\) is cyclic.

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Hint:
If \(G\) is abelian, then \(G\) is cyclic, since \(|G| = 3 \cdot 5 \cdot 17\). Now look at .
Let \(G\) have order \(p_1^{e_1} \cdots p_n^{e_n}\) and suppose that \(G\) has \(n\) Sylow \(p\)-subgroups \(P_1, \ldots, P_n\) where \(|P_i| = p_i^{e_i}\). Prove that \(G\) is isomorphic to \(P_1 \times \cdots \times P_n\).
Let \(P\) be a normal Sylow \(p\)-subgroup of \(G\). Prove that every inner automorphism of \(G\) fixes \(P\).
What is the smallest possible order of a group \(G\) such that \(G\) is nonabelian and \(|G|\) is odd? Can you find such a group?
If \(H\) is a normal subgroup of a finite group \(G\) and \(P\) is a Sylow \(p\)-subgroup of \(H\), for each \(g \in G\) show that there is an \(h\) in \(H\) such that \(gPg^{-1} = hPh^{-1}\). Also, show that if \(N\) is the normalizer of \(P\), then \(G= HN\).
Show that if the order of \(G\) is \(p^nq\), where \(p\) and \(q\) are primes and \(p>q\), then \(G\) contains a proper nontrivial normal subgroup.
Prove that the number of distinct conjugates of a subgroup \(H\) of a finite group \(G\) is \([G : N(H) ]\).

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Hint:
Define a mapping between the right cosets of \(N(H)\) in \(G\) and the conjugates of \(H\) in \(G\) by \(N(H) g \mapsto g^{-1} H g\). Prove that this map is a bijection.
Prove that a Sylow \(2\)-subgroup of \(S_5\) is isomorphic to \(D_4\).
1. Suppose \(p\) is prime and \(p\) does not divide \(m\). Show that \[\begin{aligned}\end{aligned}\].
2. Let \({\mathcal S}\) denote the set of all \(p^k\) element subsets of \(G\). Show that \(p\) does not divide \(|{\mathcal S}|\).
3. Define an action of \(G\) on \({\mathcal S}\) by left multiplication, \(aT = \{ at : t \in T \}\) for \(a \in G\) and \(T \in {\mathcal S}\). Prove that this is a group action.
4. Prove \(p \nmid | {\mathcal O}_T|\) for some \(T \in {\mathcal S}\).
5. Let \(\{ T_1, \ldots, T_u \}\) be an orbit such that \(p \nmid u\) and \(H = \{ g \in G : gT_1 = T_1 \}\). Prove that \(H\) is a subgroup of \(G\) and show that \(|G| = u |H|\).
6. Show that \(p^k\) divides \(|H|\) and \(p^k \leq |H|\).
7. Show that \(|H| = |{\mathcal O}_T| \leq p^k\); conclude that therefore \(p^k = |H|\).
Let \(G\) be a group. Prove that \(G' = \langle a b a^{-1} b^{-1} : a, b \in G \rangle\) is a normal subgroup of \(G\) and \(G/G'\) is abelian. Find an example to show that \(\{ a b a^{-1} b^{-1} : a, b \in G \}\) is not necessarily a group.

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Hint:
Let \(a G', b G' \in G/G'\). Then \((a G')( b G') = ab G' = ab(b^{-1}a^{-1}ba) G' = (abb^{-1}a^{-1})ba G' = ba G'\).
Find all simple groups \(G\) ( \(|G| \leq 60\)). Do not use the Odd Order Theorem unless you are prepared to prove it.
Find the number of distinct groups \(G\), where the order of \(G\) is \(n\) for \(n = 1, \ldots, 60\).
Find the actual groups (up to isomorphism) for each \(n\).

Symbols used here

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Prueba tu propio

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

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