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Normal Subgroups and Factor Groups: exercises
Normal Subgroups and Factor Groups: exercises — from Judson, Abstract Algebra: Theory and Applications.
Practice (14)
Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.
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For each of the following groups \(G\), determine whether \(H\) is a normal subgroup of \(G\). If \(H\) is a normal subgroup, write out a Cayley table for the factor group \(G/H\).
\(G = S_4\) and \(H = A_4\)
\(G = A_5\) and \(H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\)
\(G = S_4\) and \(H = D_4\)
\(G = Q_8\) and \(H = \{ 1, -1, I, -I \}\)
\(G = {\mathbb Z}\) and \(H = 5 {\mathbb Z}\)
Jawaby görkez
Hint:
(a) \[\begin{aligned}\end{aligned}\]
(c) \(D_4\) is not normal in \(S_4\).
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Find all the subgroups of \(D_4\). Which subgroups are normal? What are all the factor groups of \(D_4\) up to isomorphism?
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Find all the subgroups of the quaternion group, \(Q_8\). Which subgroups are normal? What are all the factor groups of \(Q_8\) up to isomorphism?
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Let \(T\) be the group of nonsingular upper triangular \(2 \times 2\) matrices with entries in \({\mathbb R}\); that is, matrices of the form \[\begin{aligned}\end{aligned}\], where \(a\), \(b\), \(c \in {\mathbb R}\) and \(ac \neq 0\). Let \(U\) consist of matrices of the form \[\begin{aligned}\end{aligned}\], where \(x \in {\mathbb R}\).
Show that \(U\) is a subgroup of \(T\).
Prove that \(U\) is abelian.
Prove that \(U\) is normal in \(T\).
Show that \(T/U\) is abelian.
Is \(T\) normal in \(GL_2( {\mathbb R})\)?
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Show that the intersection of two normal subgroups is a normal subgroup.
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If \(G\) is abelian, prove that \(G/H\) must also be abelian.
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Prove or disprove: If \(H\) is a normal subgroup of \(G\) such that \(H\) and \(G/H\) are abelian, then \(G\) is abelian.
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If \(G\) is cyclic, prove that \(G/H\) must also be cyclic.
Jawaby görkez
Hint:
If \(a \in G\) is a generator for \(G\), then \(aH\) is a generator for \(G/H\).
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Prove or disprove: If \(H\) and \(G/H\) are cyclic, then \(G\) is cyclic.
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Let \(H\) be a subgroup of index \(2\) of a group \(G\). Prove that \(H\) must be a normal subgroup of \(G\). Conclude that \(S_n\) is not simple for \(n \geq 3\).
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If a group \(G\) has exactly one subgroup \(H\) of order \(k\), prove that \(H\) is normal in \(G\).
Jawaby görkez
Hint:
For any \(g \in G\), show that the map \(i_g : G \to G\) defined by \(i_g : x \mapsto gxg^{-1}\) is an isomorphism of \(G\) with itself. Then consider \(i_g(H)\).
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Define the centralizer of an element \(g\) in a group \(G\) to be the set \[\begin{aligned}\end{aligned}\]. Show that \(C(g)\) is a subgroup of \(G\). If \(g\) generates a normal subgroup of \(G\), prove that \(C(g)\) is normal in \(G\).
Jawaby görkez
Hint:
Suppose that \(\langle g \rangle\) is normal in \(G\) and let \(y\) be an arbitrary element of \(G\). If \(x \in C(g)\), we must show that \(y x y^{-1}\) is also in \(C(g)\). Show that \((y x y^{-1}) g = g (y x y^{-1})\).
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Recall that the center of a group \(G\) is the set \[\begin{aligned}\end{aligned}\].
Calculate the center of \(S_3\).
Calculate the center of \(GL_2 ( {\mathbb R} )\).
Show that the center of any group \(G\) is a normal subgroup of \(G\).
If \(G / Z(G)\) is cyclic, show that \(G\) is abelian.
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Let \(G\) be a group and let \(G' = \langle aba^{- 1} b^{-1} \rangle\); that is, \(G'\) is the subgroup of all finite products of elements in \(G\) of the form \(aba^{-1}b^{-1}\). The subgroup \(G'\) is called the commutator subgroup of \(G\). \(G'\) commutator subgroup of \(G\)
Show that \(G'\) is a normal subgroup of \(G\).
Let \(N\) be a normal subgroup of \(G\). Prove that \(G/N\) is abelian if and only if \(N\) contains the commutator subgroup of \(G\).
Jawaby görkez
Hint:
(a) Let \(g \in G\) and \(h \in G'\). If \(h = aba^{-1}b^{-1}\), then \[\begin{aligned}ghg^{-1} & = gaba^{-1}b^{-1}g^{-1} \\ & = (gag^{-1})(gbg^{-1})(ga^{-1}g^{-1})(gb^{-1}g^{-1}) \\ & = (gag^{-1})(gbg^{-1})(gag^{-1})^{-1}(gbg^{-1})^{-1}\end{aligned}\]. We also need to show that if \(h = h_1 \cdots h_n\) with \(h_i = a_i b_i a_i^{-1} b_i^{-1}\), then \(ghg^{-1}\) is a product of elements of the same type. However, \(ghg^{-1} = g h_1 \cdots h_n g^{-1} = (gh_1g^{-1})(gh_2g^{-1}) \cdots (gh_ng^{-1})\).
Symbols used here
x belongs to A; every element of A is in B.
Inequalities that allow equality; < and > exclude it.
The two sides are different.
Naturals, integers, rationals, reals, complex numbers.
Marks the point where the statement has been established.
n divides a − b; a and b have the same remainder.
b is a multiple of a; the largest number dividing both.
A set with an operation; the do-nothing element; the element that undoes g.
Same structure; the group of cosets of a normal subgroup N.
The remainders 0…n−1 with clock arithmetic.
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.
Özüňi synla
Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.
_Ýaşa Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula