Title: Group Theory
Tags: exam2 exam1 final theorem chapter13 chapter15 final, chapter14 chapter16 chapter10 chapter13, theorem,

Q: When determining if some rule \(*\) is an operation on a set \(A\), what three properties must be satisfied?
A: 1. \(a*b\) is defined for all choices of \(a\) and \(b\) from the set \(A\).
2. \(a * b\) is an element of the set \(A\).
3. There is only one possible value for \(a * b\).

Q: An operation is commutative if [...].
In other words, an operation is commutative if [...].
A: a \(*\) b = b \(*\) a; the order of the operands doesn't matter

Q: An operation is associative if [...].
In other words, an operation is associative if [...].
A: (a \(*\) b) \(*\) c = a \(*\) (b \(*\) c); the grouping of the operands doesn't matter

Q: An element \(e\) \(\in\) \(A\) is called an [...] with respect to \(*\) if [...] for all \(a \in A\).
A: identity element; \(e * a = a * e = a\)

Q: An element \(b\) \(\in\) \(A\) is called the [...] of element \(a\) if [...]. We also write [...].
A: inverse; \(a*b=b*a=e\); \(b=a^{-1}\)

Q: When determining if some set \(G\) along with an operation \(*\) is a group, what three properties must be satisfied?
A: 1. \(*\) is associative
2. There exists an identity element \(e\) in \(G\)
3. Every element of \(G\) has an inverse in \(G\)

Q: A group whose operation is [...] is known as an [...] group.
A: commutative; abelian

Q: What are some examples of additive groups?
A: 1. Integers
2. Rational numbers
3. Real numbers.
These groups are also abelian.

Q: What are some examples of multiplicative groups?
A: 1. Positive rational numbers
2. Positive real numbers
3. Rational numbers excluding zero
4. Real numbers excluding zero
These groups are also abelian.

Q: Because groups are always [...], we can write \(abc\) to mean [...] or [...].
A: associative; \(a(bc)\); \((ab)c\)

Q: The [...] of two groups \(G\) and \(H\) is the set:
[...]
together with the operation
[...]
A: direct product; \[G \times H = \{(g,h) \mid g \in G, h \in H\} \]; \[(g_1,h_1)(g_2,h_2) = (g_1g_2,h_1h_2)\]

Q: Every group has exactly [...] [...].
A: one; identity element

Q: Every [...] in a group has exactly [...] inverse.
A: element; one

Q: Let \(a,b,c \in G\). If \(ab = ac\), then [...]. This is called the [...].
A: \(b = c\); Cancellation Law

Q: Let \(a \in G\). Then \((a^{-1})^{-1}\)=[...].
A: \(a\)

Q: Let \(a,b \in G\). Then \( (ab)^{-1}=\) [...].
A: \(b^{-1}a^{-1}\)
Extra: The order of \(a\) and \(b\) are switched.

Q: A subgroup of a group \(G\) is a subset \(H\) of \(G\) that is:
A: 1. Not empty (\(G \neq \emptyset\))
2. Closed under the group operation (if \(a,b \in H\), then \(ab \in H\))
3. Closed under inverses (if \(a \in H\), then \(a^{-1} \in H\))
For step 2, we have to prove that \(ab\) satisfies the condition of the group. For example, if the condition of the group is that \(x^3 = e\), then we must also show that for any \(x,y \in H\), \(xy \in H\), i.e. \((xy)^3 = e\).
Similarly, for step 3, if the condition of the group is that \(x^3 = e\), then we must also show that for any \(x \in H\), \(x^{-1} \in H\), i.e. \((x^{-1})^3 = e\).

Q: We call a group \(G\) a [...] group if there is an element \(a \in G\) such that
\[ G = \{ …, a ^{-1}, e, a, a^1, a^2,… \} \]
The element \(a\) is called a [...] of \(G\) and we write [...].
A: cyclic; generator; \(G = \langle a \rangle\)

A function from a set \(A\) to a set \(B\) is a rule which assigns to every element of \(A\) exactly one element of \(B\). We denote this \(f : A \rightarrow B\).
The set \(A\) is called the domain and the set \(B\) is called the codomain. The range is the set of all outputs of the function.

Q: A function \(f\) is called [...] if for all \(a,b \in A\), \(f(a) = f(b)\) implies [...]. This is also known as [...].
A: injective; \(a = b\); one-to-one
Extra: In other words, different inputs all have different outputs. If two outputs are the same, then the inputs must be the same also.

Q: A function \(f\) is called [...] if for all \(b \in B\), there exists \(a \in A\) such that [...]. This is also known as [...].
A: surjective; \(f(a) = b\); onto
Extra: In other words, every output is hit by some input. And the range equals the codomain.

Q: A function that is [...] is called [...].
A: both injective and surjective; bijective (or a bijection)

Q: If \(f : A \rightarrow B\) and \(g : B \rightarrow C\), the [...] of \(g\) with \(f\) is the function [...].
A: composition; \(g \circ f : A \rightarrow C\) defined by \([g \circ f](x) = g(f(x))\)

Q: A function \(f\) has an [...] with respect to [...] if and only if \(f\) is [...].
A: inverse; function composition; bijective
Extra: If so, the inverse function \(f^{-1}\) is defined by \(f^{-1}(y)=x\) iff \(f(x)=y\).

A permutation of a set \(A\) is a bijection from \(A\) to \(A\). The set of all permutations of \(A\), denoted \(S_A\), is a group with the operation of function composition. We call this the symmetric group on \(A\).

Q: If \(A = \{1,2,…,n\}\) for some \(n \geq 2\), we denote this group \(S_n\) and call it the [...].
We use [...] to denote the identity function defined by [...].
A: symmetric group on \(n\) elements; \(\epsilon\); \(\epsilon(x) = x\)

Q: A [...] is any [...] of a symmetric group.
A: permutation group; subgroup

Q: To determine if an operation \(x*y\) is associative, follow these steps:
A: 1. Compute \(x*(y*z)\) and \((x*y)*z\)
2. If they're equal, the operation is associative.

Q: To determine if an operation \(x*y\) is commutative, follow these steps:
A: 1. Compute \(x*y\) and \(y*x\)
2. If they're equal, the operation is commutative.

Q: To determine if every element in a set has an inverse with respect to the operation \(x*y\), follow these steps:
A: 1. Note that if there is no identity element there can be no inverse.
2. Solve \(x*x'=e\) for \(x'\).
3. If it cannot be solved, then there is no identity element.
4. If it can be solved but is dependent on a variable, then there is no identity element.
5. If it can be solved and is not dependent on a variable, then verify that \(x*x' = x'*x = e\) for all \(x\) in the set.
6. If the above holds, then \(x'\) is the inverse of \(x\).
It may be easier to prove the nonexistence of an identity element by seeing if an equation like \(0*e=0\) presents a contradiction like \(0=-1\).

Q: To determine if an operation \(x*y\) has an identity element, follow these steps:
A: 1. Solve \(x*e=x\) for \(e\).
2. If it cannot be solved, then there is no identity element.
3. If it can be solved, then verify that \(e*x = x*e = x\) for all \(x\) in the set.

Q: How many counterexamples are needed to prove that a claim is false?
A: Only one.

Q: To determine if a function \(f(x)\) is injective, follow these steps:
A: 1. Evaluate f(x).
2. Evaluate f(y).
3. If the values are not equal, then the function is injective.

If \(H\) is a subgroup of \(G\), then \(H\) itself is a group (with the same operation and identity as \(G\)) and we denote it \(H \leq G\).

Q: For all \(n \in \mathbb{N}\), [...] is the group of permutations of the set \({1,2,…,n}\) with the operation of function composition.
A: the symmetric group on \(n\) elements
Extra: The symmetric group on \(n\) elements is denoted \(S_n\).

Q: The [...] for a symmetric group is denoted \(\epsilon\) and defined by [...] for all \(x \in {1,2,…,n}\).
A: identity permutation; \(\epsilon(x) = x\)

Q: A permutation \(f \in S_n\) is called a [...] if there are \(a_1,a_2,…,a_s \in {1,2,…,n}\) such that \(f(a_1) = a_2, f(a_2) = a_3\), and so on, looping around with \(f(a_s) = a_1\).
A: cycle

Q: A permutation \(f \in S_n\) is called a [...] if there are \(a_1, a_2, \ldots, a_s \in {1,2,\ldots,n}\) such that \(f(a_1) = a_2, f(a_2) = a_2, \ldots, f(a_s) = a_1\).
Also, \(f(x) = x\) for all [...].
Finally, it has length [...].
A: cycle; \(x \in \{1,2,\ldots, n\}\) \(\setminus \{a_1, a_2, \ldots, a_s\}\); \(s\)

Q: Two cycles with no numbers in common are called [...].
A: disjoint cycles

Q: Every non-identity permutation in \(S_n\) is either a [...] or can be written as a [...].
A: cycle; product of disjoint cycles

Q: \(\begin{align*}
\left(\begin{array}{ccccccccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\
4 & 9 & 2 & 5 & 1 & 7 & 6 & 8 & 3
\end{array}\right)
\end{align*}\) =  [...]
A: \( (145)(293)(67)\)

Q: \([(137)(56)]^{-1} =\) [...]
A: \( (65)(731) = (173)(56)\)

Two groups \(G_1\) and \(G_2\) are isomorphic if there exists a bijection \(f : G_1 \rightarrow G_2\) such that \(f(ab) = f(a)f(b)\) for all \(a,b \in G_1\).
The function \(f\) is called an isomorphism from \(G_1 \rightarrow G_2\) and we write \(G_1 \cong G_2\).

A transposition is another way of writing a permutation. It has length 2. For example:
\((82673) = \) \((83)(87)(86)(82)\)
This permutation has even parity because it is written with an even number of transpositions.
Extra: A transposition is formed by taking the first element in the cycle and pairing it with the last, then the next from last, etc.
The transposition is not unique.

Q: A permutation is called [...] based on the [...] needed to write it. For example, (82673) = (83)(87)(86)(82) has [...] parity.
A: even or odd; number of transpositions; even

Q: The set of permutations in \(S_n\) forms a subgroup called [...] on \(n\) elements. This is denoted \(A_n\).
A: the alternating group

Q: An [...] \(f\) can take any of the forms
- \(f(ab) = f(a) + f(b)\)
- \(f(a + b) = f(a)f(b)\)
- \(f(a + b) = f(a) + f(b)\)
if either (or both) of the groups \(G_1\) and \(G_2\) have the operation of [...].
A: isomorphism; addition

Let \(f : G_1 \rightarrow G_2\) be an isomorphism. Then:
1. \(|G_1| =\)  \(|G_2|\)
2. \(f(e_1)\) = \(e_2\)
3. \(f(a^{-1})\) = \([f(a)]^{-1}\)  for all \(a \in G_1\)
4. If \(G_1 = \langle a \rangle\) then \(G_2 = \langle f(a) \rangle\)

Q: What are some clues that two groups are not isomorphic?
A: 1. \(G_1\) is commutative and \(G_2\) is not.
2. \(G_1\) has an element which is its own inverse and \(G_2\) does not.
3. \(G_1\) has an element of order \(x\) but \(G_2\) does not.
4. \(G_1\) and \(G_2\) have different numbers of elements.

Q: Let \(a\) be an element of a group \(G\). If there exists a non-zero \(m \in \mathbb{Z}\) such that \(a^m = e\), then the [...] of \(a\) in \(G\) is the [...] \(n\) such that \(a^n = e\).
A: order; least positive integer
Extra: We denote this by ord\((a) = n\).

Q: If there does not exist a non-zero \(m\) such that \(a^m = e\), then we say that \(a\) has order [...].
A: infinity
Extra: This is denoted ord\((a) = \infty\).

Q: If the operation in a set \(G\) is addition, we define the [...] of \(a \in G\) to be the [...] \(n\) such that [...].
A: order; least positive integer; \(na = 0\)

Q: The identity element always has order [...], and is the [...].
A: 1; only element with order 1

Q: Every non-zero element of \(\mathbb{Z}\) has order [...].
A: infinity

Q: In \(\mathbb{Z}_n\), ord(1) = [...].
A: \(n\)
Extra: This means that we must add 1 \(n\) times to produce \(e\) (i.e., 0).

Q: In \(S_n\), the order of a cycle is equal to its [...]. For example, \((2534) \in S_5\) has [...].
A: length; order 4

Q: In \(S_n\) the order of a product of disjoint cycles is the [...] of the [...].
A: least common multiple; cycle lengths

Q: \((15)(238)(4679) \in S_9\) has order [...], because that's the [...] of [...].
A: 12; least common multiple; 2, 3, and 4

Q: In a direct product \(G \times H\) , the order of \((g, h)\) is the [...] of the order of \(g\) in \(G\) and the order of \(h\) in \(H\).
A: least common multiple
Extra: Like the least common multiple of the least common multiples. For example, the order of \((8, 3)\) in \(\mathbb{Z}_{10} \times \mathbb{Z}_6\) is lcm\((5, 2) = 10\).

Q: The order of \((8, 3)\) in \(\mathbb{Z}_{10} \times \mathbb{Z}_6\) is [...].
A: lcm\((5, 2) = 10\)

Q: If ord\((a) = n < \infty\), then the elements \(a^0, a, a^2,…, a^{n-1}\) are [...] and every power of \(a\) is [...].
A: distinct; equal to one of these elements

Q: If the order of \(a\) is [...], then \(a^t = e\) if and only if \(t\) is a [...] of \(n\).
A: finite; multiple
Extra: In other words, the only powers of \(a\) that equal the identity are \(a^0, a^{\pm1n}, a^{\pm2n}, …\).

Q: A group \(G\) is [...] if there exists an element \(a \in G\) such that \(G = \{\ldots, a^{-2}, a^{-1}, a^0, a, a^2, \ldots\}\).
We use \(\langle a \rangle\) to denote this [...] group and call the element \(a\) a [...].
A: cyclic; cyclic; generator

Q: If \(a\) has finite order, then \(\langle a \rangle\) has order \(n\) and is isomorphic to [...].
The function [...] is an [...].
A: \(\mathbb{Z}_n\); \(f(a^i) = i\); isomorphism

Q: If \(a\) has infinite order, then \(\langle a \rangle\) is an infinite group and is [...] to \(\mathbb{Z}\).
The function [...] is an [...].
A: isomorphic; \(f(a^i) = i\); isomorphism

Q: The order of every element in \(\mathbb{Z}_n\) [...] \(n\).
A: divides

Q: Every subgroup of \(\mathbb{Z}_n\) is [...].
A: cyclic

Q: \(\mathbb{Z}_n\) has exactly one subgroup of [...] for every [...].
A: order \(k\); positive divisor \(k\) of \(n\)

Q: Let \(k \gt 1\) be a positive divisor of \(n\). The subgroup \(\mathbb{Z}_n\) of order \(k\) is generated by [...].
A: \(\frac{n}{k}\)

A partition of a set A is a family of nonempty subsets which are mutually disjoint and whose union is all of A.

Q: If two classes of a partition contain the same element, then they are [...].
A: the same class

Q: When partitioning a set \(A\), [...] element \(x\) of \(A\) lies in one of the partition's [...].
A: every; classes

Q: An equivalence relation ~ is:
- [...]
- [...]
- [...]
A: Reflexive (every element equals itself: \(a \sim a\)); Symmetric (order doesn't matter: \(c \sim d\) → \(d \sim c\)); Transitive (\(a \sim b\) and \(b \sim c\) → \(a \sim c\))

Q: The set of all elements [...] to \(x\) is called the [...] of \(x\) and is denoted [...].
A: equivalent; equivalence class; [\(x\)]

Q: If x ~ y, then [...].
A: [x] = [y]
Extra: In other words, if two elements are equivalent, they have the same equivalence class.

Q: If ~ is an equivalence relation on \(A\), the family of [...] is [...].
A: all the equivalence classes; a partition of \(A\)

Q: What is a homomorphism?
A: A homomorphism is a translation from one group to another which preserves the operation of the group. Like an isomorphism, but without the requirement of being a bijection.
Formally: A function \(f\) which maps elements from a group \(G\) to a group \(H\) preserving the group operation \(f(ab) = f(a)f(b)\).

Q: A homomorphism maps the identity element in \(G\) to [...].
A: the identity element in \(H\)
Extra: In other words, \(f(e_G) = e_H\).

Q: What is a normal subgroup?
A: A subgroup is normal if it is closed under conjugation. If the parent group is abelian, then all subgroups are normal.
Formally: A subgroup \(H\) of \(G\) for which \(ghg^{-1} \in H\) for all \(g \in G\) and \(h \in H\).

Q: The [...] Theorem:
[...]
In other words, [...].
A: Fundamental Homomorphism; Let \(f: G \rightarrow H\) be a surjective homomorphism and let \(K\) be the kernel of \(f\), then: \(G/K \cong H\).; every homomorphic image of \(G\) is isomorphic to a quotient group of \(G\). Specifically, the quotient group of \(G\) by the kernel of \(f\)

Q: What is a homomorphic image?
A: If there exists a surjective homomorphism from \(G\) onto \(H\), then \(H\) is a homomorphic image of \(G\).

Q: Theorem [...]:
Let [...].
In other words, [...].
A: 16.1; \(f : G \rightarrow H\) be a homomorphism with kernel \(K\). Then \(f(a) = f(b)\) iff \(Ka = Kb\); any two elements \(a\) and \(b\) in \(G\) have the same image under \(f\) iff they are in the same coset of \(K\)

Q: Theorem [...]:
[...]
In other words, [...].
A: 15.1; If \(H\) is a normal subgroup of \(G\), then \(aH=Ha\) for every \(a \in G\).; there is no distinction between left and right cosets for a normal subgroup

Q: Theorem [...]:
[...]
In other words, [...].
A: 15.2; Let \(H\) be a normal subgroup of \(G\). If \(Ha = Hc\) and \(Hb = Hd\), then \(H(ab) = H(cd)\).; coset multiplication is a valid operation on normal subgroups. In addition, \(G/H\) with coset multiplication is a group

Q: Theorem [...]:
[...]
A: 15.3; The quotient group \(G/H\) with coset multiplication is a group.

Q: Theorem [...]:
[...]
In other words, [...].
A: 15.4; The quotient group \(G/H\) is a homomorphic image of \(G\).; when we construct quotient groups of \(G\), we are constructing homomorphic images of \(G\). A homomorphic image is a surjective homomorphism

Theorem 15.5:
Let \(G\) be a group and \(H\) a subgroup of \(G\). Then:
(i) \(Ha = Hb\) iff \(ab^{-1} \in H\) and
(ii) \(Ha = H\) iff \(a \in H\)

In other words, cosets are equal iff an element in one and an inverse in another are in \(H\). 
And, a coset is equal to the identity \(H\) iff the representative of that coset is in \(H\).

Q: Theorem [...]:
[...]
In other words, [...]
A: 14.1; Let \(G\) and \(H\) be groups, and let \(f : G \rightarrow H\) be a homomorphism. Then:
(i) \(f(e_G) = e_H\) and
(ii) \(f(a^{-1}) = [f(a)]^{-1}\) for every element \(a \in G\); the homomorphism \(f\) maps the identity in \(G\) to the identity in \(H\) and pairs an inverse in G to its inverse in H.

Q: What is a kernel?
A: Let \(f : G \rightarrow H\) be a homomorphism. Then its kernel \(K = \{x \in G \mid f(x) = e\}\).
In other words, the kernel of \(f\) is the set \(K\) of all the elements of \(G\) which are carried by \(f\) onto the identity element of H.

Q: Theorem [...]:
Let [...]
A: 14.2; \(f : G \rightarrow H\) be a homomorphism. Then:
(i) The kernel of \(f\) is a normal subgroup of \(G\), and
(ii) The range of \(f\) is a subgroup of \(H\)

Q: Theorem [...]:
Let [...]
In other words, [...].
A: 13.1; \(G\) be a group and let \(H\) be a fixed subgroup of \(G\). The family of all the cosets \(Ha\), as a ranges over \(G\), is a partition of \(G\).; all the cosets of \(H\) together are equal to \(G\). Nothing is left out

Q: Theorem [...]:
[...]
In other words, [...]
A: 13.2; If \(Ha\) is any coset of \(H\), there is a one-to-one correspondence from \(H\) to \(Ha\).; any coset \(Ha\) has the same number of elements as \(H\), and therefore all the cosets have the same number of elements.

Q: [...]:
[...]
In other words, [...].
A: Lagrange's Theorem; Let \(G\) be a finite group, and \(H\) be any subgroup of \(G\). The order of \(G\) is a multiple of the order of \(H\).; the order of any subgroup \(H\) of a group \(G\) is a divisor of the order of \(G\)

Q: Theorem [...]:
[...]
Finally, [...].
A: 13.4; If \(G\) is a group with a prime number \(p\) of elements, then G is a cyclic group. Furthermore, any element \(a \neq e\) in \(G\) is generator of \(G\).; there is only one group (as far as isomorphism is concerned) of any given prime order \(p\)

Q: Theorem [...]:
[...].
A: 13.5; The order of any element of a finite group divides the order of the group

Q: What is the index of a subgroup?
A: The index of a subgroup \(H\) in \(G\) is the number of cosets of \(H\) in \(G\). It is denoted:
\([G : H] = \frac{\mid G \mid}{\mid H \mid}\)
For example, let \(G\) be the additive group of integers modulo 12, and let \(H\) be the subgroup generated by 4, so \(H = \{0, 4, 8\}\). Then there are four distinct cosets of \(H\), namely:
\(H = \{0, 4, 8\} \)
\(1+H = \{1, 5, 9\}\)
\(2+H = \{2, 6, 10\}\)
\(3+H = \{3, 7, 11\}\)
In other words, the cosets are \(H\) and its translations \(1+H\), \(2+H\), and \(3+H\).
Together these four cosets contain all the elements of \(G\), and \(|G:H| = \frac{|G|}{|H|} = \frac{12}{3} = 4\).

Q: Theorem [...]:
[...]
In other words, \(n\) will divide all \(t\) such that \(a^t = e\).
A: 10.5; Suppose an element \(a\) in a group has order \(n\). Then \(a^t = e\) iff \(t\) is a multiple of \(n\).

Q: If a function \(f\) is a [...], then its [...] is a [...].
A: homomorphism; kernel ker\((f)\); normal subgroup

By Lagrange's Theorem, if a group has prime order \(p\), then all subgroups of that group must have order \(p\) or order 1 (the trivial subgroup containing only the identity).

A homomorphism \(f : G \rightarrow H\) is injective iff ker(\(f\)) = \(\{e\}\).
In other words, if the only element in \(G\) mapped to the identity of \(H\) is \(e_G\), then the function \(f\) is injective. The converse is also true.

Q: [...] Theorem:
If [...], then [...].
A: Cauchy's; \(G\) is a finite group, and \(p\) is a prime divisor of the order of \(G\); \(G\) has an element of order \(p\)

The center of a group \(G\) is the set of elements that commute with every element of \(G\). It is denoted Z(\(G\)).

Q: [...] implies that [...].
In other words, [...].
A: \(aH = Ha\); every element in \(aH\) is in \(Ha\) and conversely; for any \(h \in H\), there is some \(k \in H\) such that \(ah = ka\) and there is some \(l \in H\) such that \(ha = al\)

Q: If \(G\) is a group and there exists an element \(a \in G\) such that ord\((a) = |G|\), then \(G\) is [...] and [...] is its generator.
In other words, [...].
A: cyclic; \(a\); \(G = \langle a \rangle\)