Quiz
Abstract Algebra THEOREMS
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Questions in this quiz (24)
1Let R be an equivalence relation defined on a set S and define [a]= {b in S| bRa}. Then _______________
- A{[a]|a is in S} is a partition of S
- B{[a]|a is in S} is a subset of S
- C{[a]|a is in S} is a subgroup of S
2True or False, if S is a set closed under a binary operation * , then S contains more than one identity
- ATrue
- BFalse
3Let S be a set closed under a binary operation *, with identity e, where * is ([blank_start]Answer 1[blank_end]). Then x (in S) has ([blank_start]Answer 2[blank_end]) Inverses.
- AClosed
- BCommutative
- CAssociative
- DDistributive
- EAt most 1
- FMore than 1
4Let S be a set, closed under an associative binary operation *, with identity e. Then, if x^-1 and y^-1 exist then
- A(x*y)^-1 = y^-1 * x^-1
- B(x*y)^-1 =x^-1 * y^-1
5Suppose n is a Irrational Number and S a set closed under associative binary operation * then x^n = x * x * x * ... * x (n terms)
- ATrue
- BFalse
6A group is a pair (S,*) where S is a non-empty set and * a binary operation on S such that;
S is [blank_start]ANSWER 1[blank_end] under *,
* is [blank_start]ANSWER 2[blank_end] on S,
S has [blank_start]ANSWER 3[blank_end] w.r.t. to * denoted e,
every element in S has an [blank_start]ANSWER 4[blank_end], denoted x^-1 in S- AClosed
- BIsometric
- CEquivalent
- DPrime
- EAssociative
- FDistributive
- GNon-Zero
- HNon-Empty
- IField
- JCommutativity
- KDelta
- LIdentity
- MInverse
- NElement
- OEven parity
- POdd power
7Tick the properties required for the General Linear Group of degree n
- ANon-singular (Invertible)
- BMatrix size n x n
- CSymmetric matrix
- DTransposition notation
8What are the properties of an ISOMETRY
- ABijective Mapping
- BDistance Preserved
- Cf(xy) = f(x)f(y)
- DSymmetrical
9If P is the set of points representing a figure in space, and G is the set of all isometries that map P onto itself, then (G, *) is a group called the Symmetry Group Of P
- ATrue
- BFalse
10What does the following mean in words?
- AThe set S has four elements in it.
11Let (G, *) be a group with identity element e and consider some element a in G. What does it mean to say that a has period w?
- Aa^w = e
- Ba^w = 0
12Complete the following definition: Let (G,*) be a group and H be a [blank_start]subset[blank_end] of G. Then, H is a [blank_start]subgroup[blank_end] of G [blank_start]if and only if[blank_end] (H,*) is a group.
- Asubset
- Bgroup
- Cunit
- Disometry
- Esubgroup
- Fgroup
- Gset
- Hperiod
- Iif and only if
- Jif
- Kwhen
- Lproviding
13True or False: If (H,*) is a subgroup of (G,*) then;
i) they have the same identity
ii) the inverse of x in H is the inverse of x in G- ATrue
- BFalse
14Complete the definition: If H is a [blank_start]non-empty[blank_end], finite, subset of the [blank_start]elements[blank_end] of a group (G,*), then (H,*) is a subgroup of (G,*) [blank_start]if and only if[blank_end] H is [blank_start]closed[blank_end] under *.
- Anon-empty
- Bnon-zero
- Cring
- Dmatrix
- Eelements
- Fperiod
- Gorder
- Hparity
- Iif and only if
- Jif
- Kand
- Lwhen
- Mclosed
- Ncommutative
- Oassociative
- Pinvertible
15Let (G,*) be a group and x be in G. Then _____________ forms a subgroup of (G,*) called the cyclic subgroup generated by x, denoted <x>
- A{x^n|n is an integer}
- B{yx|y is in a group H}
- C{x+kn|n,k are integers}
- Dfi(n) = |{x is natural| x<_ n and gcd (x,n)=1}| for all naturals n
16Select the properties for which a binary operation is uniquely defined
- AClosed
- BAssociativity
- CCommutativity
- DIdentity
- EDistributivity
- FRelation
- GInverse
- HFinite
17True or False, if (G,*) is a cyclic group with generator a, denoted <a>. Then |G| = period of a
- ATrue
- BFalse
18Which ONE of the following is used to describe a LEFT COSET?
- A{x^n|n is an integer}
- Bfi(n) = |{x is natural| x<_ n and gcd (x,n)=1}| for all naturals n
- C{x+kn|n,k are integers}
- D{xy|y is in a group H}
19Which of the following describes that
a) the period of the coset is the period of the set
b) each element can only be in one coset
( G is a finite group and H a subgroup). SELECT TWO- AFor all g in G, |gH| = |H|
- BFor all g1, g2 in G, either g1H = g2H or g1H n g2H = EmptySet
- C|gH| = 0
- Dg1H = g2H and g1H n g2H = g1H u g2H
20[blank_start]LAGRANGE[blank_end]'S THEOREM: Let (H,*) be a subgroup of a [blank_start]finite[blank_end] group (G,*). Then the number of [blank_start]elements[blank_end] in H divides the number of elements in G; that is IHI | IGI
- ALagrange
- BCauchy
- CSilow
- DWilson
- Efinite
- Finfinite
- Gnon-empty
- Habelian
- Ielements
- Jsubgroups
- Ksets
- Lidentities
21Let (G,*) be a finite group of order n and let g be an element with period k. Then i) k|n (ie. the period of an element divides the order or the group) and ii) g^n = e
- ATrue
- BFalse
22Complete the theorem: Let (G,*) be a [blank_start]finite[blank_end] group of order p, where p is [blank_start]prime[blank_end]. Then, (G,*) is [blank_start]cyclic[blank_end]
- Afinite
- Binfinite
- Cclosed
- Dassociative
- Eprime
- Fnon-empty
- Gnon-zero
- Hirrational
- Icyclic
- Jempty
- Ka field
- Lpartitioning
23Which of the following are properties of a homomorphism?
- Af(xy)=f(x)of(y)
- Bbijective
- Cdistances preserved
24Which of the following hold for an Isomorphism of groups f: G1 -> G2
- Aidentity in G1 = identity in G2
- B|x| = |f(x)| same period
- CG1 is abelian iff G2 is abelian
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- Mathematics Quiz on Abstract Algebra THEOREMS , created by Jess :) on 16/06/2017.
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- Jess :)
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