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Cosets and Lagrange's Theorem: exercises

Cosets and Lagrange's Theorem: exercises — from Judson, Abstract Algebra: Theory and Applications.

Practice (23)

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

Suppose that \(G\) is a finite group with an element \(g\) of order \(5\) and an element \(h\) of order \(7\). Why must \(|G| \geq 35\)?

Giải đáp

Hint:
The order of \(g\) and the order \(h\) must both divide the order of \(G\).
Suppose that \(G\) is a finite group with \(60\) elements. What are the orders of possible subgroups of \(G\)?

Giải đáp

Hint:
The possible orders must divide \(60\).
Prove or disprove: Every subgroup of the integers has finite index.

Giải đáp

Hint:
This is true for every proper nontrivial subgroup.
Prove or disprove: Every subgroup of the integers has finite order.

Giải đáp

Hint:
False.
List the left and right cosets of the subgroups in each of the following.
1. \(\langle 8 \rangle\) in \({\mathbb Z}_{24}\)
2. \(\langle 3 \rangle\) in \(U(8)\)
3. \(3 {\mathbb Z}\) in \({\mathbb Z}\)
4. \(A_4\) in \(S_4\)
5. \(A_n\) in \(S_n\)
6. \(D_4\) in \(S_4\)
7. \({\mathbb T}\) in \({\mathbb C}^\ast\)
8. \(H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\) in \(S_4\)
Giải đáp

Hint:
(a) \(\langle 8 \rangle\), \(1 + \langle 8 \rangle\), \(2 + \langle 8 \rangle\), \(3 + \langle 8 \rangle\), \(4 + \langle 8 \rangle\), \(5 + \langle 8 \rangle\), \(6 + \langle 8 \rangle\), and \(7 + \langle 8 \rangle\); (c) \(3 {\mathbb Z}\), \(1 + 3 {\mathbb Z}\), and \(2 + 3 {\mathbb Z}\).
Describe the left cosets of \(SL_2( {\mathbb R} )\) in \(GL_2( {\mathbb R})\). What is the index of \(SL_2( {\mathbb R} )\) in \(GL_2( {\mathbb R})\)?
Verify Euler's Theorem for \(n = 15\) and \(a = 4\).

Giải đáp

Hint:
\(4^{\phi(15)} \equiv 4^8 \equiv 1 \pmod{15}\).
Use Fermat's Little Theorem to show that if \(p = 4n + 3\) is prime, there is no solution to the equation \(x^2 \equiv -1 \pmod{p}\).
Show that the integers have infinite index in the additive group of rational numbers.
Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.
Let \(H\) be a subgroup of a group \(G\) and suppose that \(g_1, g_2 \in G\). Prove that the following conditions are equivalent.
1. \(g_1 H = g_2 H\)
2. \(H g_1^{-1} = H g_2^{-1}\)
3. \(g_1 H \subset g_2 H\)
4. \(g_2 \in g_1 H\)
5. \(g_1^{-1} g_2 \in H\)
If \(ghg^{-1} \in H\) for all \(g \in G\) and \(h \in H\), show that right cosets are identical to left cosets. That is, show that \(gH = Hg\) for all \(g \in G\).

Giải đáp

Hint:
Let \(g_1 \in gH\). Show that \(g_1 \in Hg\) and thus \(gH \subset Hg\).
What fails in the proof of if \(\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H\) is defined by \(\phi( gH ) = Hg\)?
Suppose that \(g^n = e\). Show that the order of \(g\) divides \(n\).
The cycle structure of a permutation \(\sigma\) is defined as the unordered list of the sizes of the cycles in the cycle decomposition \(\sigma\). For example, the permutation \(\sigma = (1 \, 2)(3 \, 4 \, 5)(7 \, 8)(9)\) has cycle structure \((2,3,2,1)\) which can also be written as \((1, 2, 2, 3)\).
Show that any two permutations \(\alpha, \beta \in S_n\) have the same cycle structure if and only if there exists a permutation \(\gamma\) such that \(\beta = \gamma \alpha \gamma^{-1}\). If \(\beta = \gamma \alpha \gamma^{-1}\) for some \(\gamma \in S_n\), then \(\alpha\) and \(\beta\) are conjugate.
If \(|G| = 2n\), prove that the number of elements of order \(2\) is odd. Use this result to show that \(G\) must contain a subgroup of order \(2\).
Suppose that \([G : H] = 2\). If \(a\) and \(b\) are not in \(H\), show that \(ab \in H\).
If \([G : H] = 2\), prove that \(gH = Hg\) for all \(g \in G\).
Let \(H\) and \(K\) be subgroups of a group \(G\). Prove that \(gH \cap gK\) is a coset of \(H \cap K\) in \(G\).

Giải đáp

Hint:
Show that \(g(H \cap K) = gH \cap gK\).
Let \(H\) and \(K\) be subgroups of a group \(G\). Define a relation \(\sim\) on \(G\) by \(a \sim b\) if there exists an \(h \in H\) and a \(k \in K\) such that \(hak = b\). Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets. Compute the double cosets of \(H = \{ (1),(1 \, 2 \, 3), (1 \, 3 \, 2) \}\) in \(A_4\).
Let \(G\) be a cyclic group of order \(n\). Show that there are exactly \(\phi(n)\) generators for \(G\).
Let \(n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}\), where \(p_1, p_2, \ldots, p_k\) are distinct primes. Prove that \[\begin{aligned}\end{aligned}\].

Giải đáp

Hint:
If \(\gcd(m,n) = 1\), then \(\phi(mn) = \phi(m)\phi(n)\) ( in ).
Show that \[\begin{aligned}\end{aligned}\] for all positive integers \(n\).

Symbols used here

a \equiv b \pmod n

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

x \in A,\ A \subseteq B

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

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\sigma,\ s,\ \sigma^2

standard deviation, sample s.d., variance
Typical distance from the mean; its square.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\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 \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.

Thử đi.

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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