Flashcards
Math 225 Abstract Algebra
Cards in this deck (25)
- 1Define a function f: X → YLet X and Y be non-empty sets. A function
f: X → Y is a subset Γ ⊆ X × Y such that for all x ∈ X there exists a unique y ∈ Y
such that (x, y) ∈ Γ. - 2Define a binary operationLet S be a non-empty set. A binary operation on S is a
function ∗: S × S → S. - 3Define an associative binary operationLet S be a non-empty set and ∗: S × S → S be a binary
operation on S. We say ∗ is associative if
(a ∗ b) ∗ c = a ∗ (b ∗ c)
for all a, b, c ∈ S. - 4Define what it means for a binary operation to have an identityLet S be a non-empty set and let ∗: S ×S → S be a binary
operation on S. We say that an element e ∈ S is an identity for ∗ if
e ∗ a = a = a ∗ e
for all a ∈ S. If such an element e exists, we say that ∗ has an identity. - 5Define what it means for a binary operation to be commutativeLet S be a non-empty set and let ∗: S ×S → S be a binary
operation on S. We say that ∗ is commutative if
a ∗ b = b ∗ a
for all a, b ∈ S. - 6Define what it means for an element a ∈ S to have an inverseLet S be a non-empty set and ∗: S × S → S a binary
operation on S. Assume that there exists e ∈ S such that e is an identity
for ∗. We say that an element a ∈ S has an inverse with respect to ∗ if there
exists b ∈ S such that
a ∗ b = e = b ∗ a. - 7Define a groupA group is an algebraic structure (G, ∗) with ∗ an associative
binary operation, having an identity and such that every element of G
has an inverse with respect to ∗. - 8Define an Abelian groupA group (G, ∗) is called Abelian if ∗ is commutative.
- 9Define a ringA ring is an algebraic structure (R,+,×) such that (R, +)
is an Abelian group, × is associative and we have
a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
for all a, b, c ∈ R. The identity element for the group (R, +) will be denoted
0R. - 10Define a fieldA field is an algebraic structure (F,+,×) such that (F,+,×)
is a ring, × is commutative and has an identity 1_F ≠ 0_F and every element
of F \ {0_F } has an inverse with respect to ×. - 11Define what it means for a group to be finiteA group (G, ∗) is called finite if the set G is finite.
- 12What is the order of a groupThe order of a group (G, ∗) is the cardinality of G, |G|.
- 13Define the symmetric group of degree nThe symmetric group of degree n is defined to be the group
of bijections (Bij({1, 2, . . . , n}), ◦) and is denoted S_n. - 14Define a cycleLet r ≤ n and let a1, a2, . . . , ar be a list of distinct elements
of {1, 2, . . . , n}. Let π ∈ S_n be the permutation defined by
π(ai) = ai+1 for 1 ≤ i ≤ r − 1
π(ar) = a1
π(a) = a for a /∈ {a1, a2, . . . , ar}
We will call π a cycle of length r, or an r-cycle, and we will write (a1 a2 · · · ar). - 15Define a disjoint cycleLet π = (a1 a2 · · · ar) and σ = (b1 b2 · · · bs) be cycles in
S_n. If the sets {a1, a2, . . . , ar} and {b1, b2, . . . , bs} are disjoint, we say that π
and σ are disjoint cycles. This terminology is extended in the obvious way
to sets of more than two cycles. - 16Every permutation in S_n can be written as a product of disjoint cycles; moreover this expression is unique up to: i) ii) iii) (This is a theorem you must remember. This is proved on page 26)(i) the order in which the cycles occur,
(ii) the different ways of writing each cycle (rotating the cycle), and
(iii) the presence or absence of 1-cycles. - 17Define what it means for a permutation to be in disjoint cycle notationA permutation which is written as a product of disjoint
cycles is said to be in disjoint cycle notation; if all 1-cycles are included, it
is said to be in full cycle notation. - 18Define the order of a permutation πLet π ∈ S_n. The order o(π) of π is the minimal r ∈ N such that
π^r = π π · · ·π = ι (identity). (r times). - 19Proof If π = γ1γ2 · · · γk, where the γi are disjoint cycles, then the order of π is the lowest common multiple of the lengths of the cycles γi.Proof. Since the γi commute, for n ∈ N we have
πn = (γ1 · · · γk)n = γ1
n · · · γk
n.
As the γi move different elements, πn = ι if and only if γi
n = ι for all i.
Since by remark (e) above, the order of each cycle γi is its length, the result
follows. - 20Define a transpositionA 2-cycle is called a transposition.
- 21Proof If n > 1, every permutation in S_n can be written as a product of transpositions.Proof. By Theorem 3.7 it suffices to show that every r-cycle for r > 1 can
be written as a product of transpositions. We may write
(a_1 a_2 a_3 . . . a_(r−1) a_r) = (a_1 a_2)(a_2 a_3) · · · (a_(r−1) a_r).
For looking at the product of transpositions and remembering that we read
right to left, we see that a_1 is fixed by all the 2-cycles until (a_1 a_2), which
sends a_1 to a_2. Then a_2 is fixed by all the 2-cycles until (a_2 a_3), so we
have “(a_1 a_2 a_3 ” so far. Continuing in this way, we see that the product of
transpositions is equal to the r-cycle as claimed. - 22Define the sign of a permutationGiven π ∈ S_n, the sign of π is defined by
sign(π) = (−1)^(ν(π)). - 23Let π ∈ S_n and let τ ∈ S_n be a transposition. Then what is sign(πτ )sign(πτ ) = (−1) · sign(π) = −sign(π).
- 24Proof Let π, σ ∈ S_n. Then sign(π) sign(σ) = sign(πσ).By Theorem 3.14 we may write
π = τ_1 · · · τ_r, σ = τ_1' · · · τ_s'
for some transpositions τ_1, . . . , τ_r, τ_1', . . . , τ_s'. Then πσ = τ_1 · · · τ_rτ_1' · · · τ_s'
and so
sign(π) sign(σ) = (−1)^(r)(−1)^(s) = (−1)^(r+s) = sign(πσ). - 25What is the group denoted A_n known as?The subgroup of S_n consisting of the even permutations
is called the alternating group of degree n and is denoted A_n.
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- Flashcards on definitions and important theorems within the module Math 225
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- James Middleton
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