Group Theory Concepts Quiz

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Questions and Answers

What is the result of the equation $o(x) = p_i m_i$ when $m_i = 0$?

$o(x) = 0$

$o(x) = 1$ (correct)

$o(x) = p_i$

$o(x) = p_i m_i$

When $x$ is an element of $H_i$, then $o(x)$ is equal to the order of $H_i$.

True

What does $H_i igcap H_1 H_2 eq ext{e}$ imply?

It suggests that their intersection contains elements other than the identity element.

In the expression $x = (h_1 h_2 ... h_{i-1} h_{i+1} ... h_r)$, $h_t$ is the _____ element.

identity Signup and view all the answers

Match the symbols with their meanings:

$o(x)$ = Order of element x $e$ = Identity element $H_i$ = Subgroup containing element x $m_i$ = Multiplicity of pi in factorization Signup and view all the answers

Which of the following statements is true regarding the identity element in a group?

The identity element is always unique. Signup and view all the answers

In an Abelian group, the operation is commutative.

True Signup and view all the answers

What is a subgroup of a group G?

A non-empty subset H of G that is itself a group with the same binary operation. Signup and view all the answers

A group with infinite elements is called an __________ group.

infinite Signup and view all the answers

Match each term with its corresponding description:

Normal Subgroup = Na = aN for all a in G Proper Subgroup = A subgroup that is not equal to G or {e} Coset = Set formed by Ha = {ha : h in H} Finite Group = A group with a limited number of elements Signup and view all the answers

Which statement is true regarding the cancellation laws in group theory?

If ab = ac, then b = c. Signup and view all the answers

All elements in a group must have unique inverses.

True Signup and view all the answers

The sets Ha and aH are called __________ of a subgroup H in G.

cosets Signup and view all the answers

What is the relationship between the orders of the groups HxK and HxKx−1?

o(HxK) = o(HxKx−1) Signup and view all the answers

If K is a subgroup of G, then xKx−1 is not a subgroup of G.

False Signup and view all the answers

What is the highest power of p such that $p^n$ divides the order of G?

n Signup and view all the answers

According to Lagrange’s Theorem, o(H) o(H ∩ xKx−1) = ______.

o(H ∩ xKx−1) Signup and view all the answers

Match the following terms with their definitions:

H = A Sylow p-subgroup of G K = Another Sylow p-subgroup of G n = The highest power of p dividing the order of G o(H) = The order of the subgroup H Signup and view all the answers

What can be inferred if H and K are two Sylow p-subgroups of G?

H and K are conjugate in G Signup and view all the answers

If o(H) = o(K) = p^n, then H and K must be equal.

False Signup and view all the answers

What does the notation $xKx^{-1}$ signify in group theory?

Conjugation of subgroup K by element x Signup and view all the answers

What condition must hold for a subgroup to exist in G when considering the order o(G')?

o(G') must be less than o(G) and pk divides o(G') Signup and view all the answers

If pm does not divide the order of any proper subgroup H of G, then pm divides o(G).

True Signup and view all the answers

What theorem is applied to deduce the existence of an element a in Z(G) such that a^p = e?

Cauchy Theorem for finite abelian groups Signup and view all the answers

The cyclic subgroup generated by an element a is denoted as K = < a > = {a, a 2 , a 3 ,..., a ______}

a^p Signup and view all the answers

If $o(G) = pq$, where $p$ and $q$ are distinct primes and $p < q$, what can be deduced about the structure of $G$?

G is cyclic. Signup and view all the answers

Match the following concepts with their definitions:

o(G) = Order of group G Z(G) = Center of group G K = Cyclic subgroup generated by element a N(a) = Normal subgroup containing element a Signup and view all the answers

The normalizer $N(H igcap K)$ is equal to the group $G$ if and only if $H$ is normal in $G$.

True Signup and view all the answers

What does the class-equation for G describe?

The relationship between the order of G and its center Signup and view all the answers

What is the condition for $N(H igcap K)$ to equal the entire group $G$?

o(N(H ∩ K)) = o(G) Signup and view all the answers

If $o(N(H igcap K)) = 108$, it follows that $G$ is __________.

not simple Signup and view all the answers

Every subgroup of Z(G) is a normal subgroup of G.

True Signup and view all the answers

Match the Sylow theorem notations with their meanings:

$n_p$ = Number of Sylow p-subgroups $n_q$ = Number of Sylow q-subgroups $H$ = Unique Sylow p-subgroup $K$ = Unique Sylow q-subgroup Signup and view all the answers

In case I, what is assumed about the order of a subgroup H of G?

pm divides o(H) Signup and view all the answers

What condition leads to a contradiction when analyzing the number of Sylow $p$-subgroups?

If $1 + kp = q$. Signup and view all the answers

The Sylow third theorem implies that a group with a unique Sylow subgroup is guaranteed to be abelian.

False Signup and view all the answers

What can be concluded if $o(H) = p$ and $H$ is unique in $G$?

H is normal in G. Signup and view all the answers

Which condition must be satisfied for G to be considered an internal direct product of its Sylow subgroups?

The intersection of any subgroup with the product of the others must be the identity. Signup and view all the answers

Every subgroup of an abelian group is normal in that group.

True Signup and view all the answers

What is the identity element denoted as in group theory?

e Signup and view all the answers

If o(H_i) = p_i^{n_i}, then the order of the subgroup H_i is a power of the prime number _______.

p_i Signup and view all the answers

Match the concepts with their definitions:

Normal subgroup = A subgroup that is invariant under conjugation by any element of the group Sylow subgroup = A maximal p-subgroup of a group Internal direct product = A product of subgroups where each pair of distinct subgroups intersects trivially Abelian group = A group where the group operation is commutative Signup and view all the answers

In proving G is an internal direct product, which of the following is not required?

G must contain exactly one Sylow subgroup for each prime factor. Signup and view all the answers

The equation x = h1h2...hi-1hi+1...hr implies that x is an element of Hi for any i.

False Signup and view all the answers

What is the significance of the notation P_j for j ≠ i in the context of Sylow subgroups?

It indicates that the subgroup order is a power of the prime number p_j. Signup and view all the answers

Study Notes

Course Information

Course: Abstract Algebra
Paper Code: 20MAT21C1
Semester: 1
University: Maharshi Dayanand University, Rohtak

Course Outcomes

Students will be able to apply group theoretic reasoning to group actions.
Students will learn properties and analysis of solvable and nilpotent groups, Noetherian and Artinian modules and rings.
Students will apply Sylow's theorems to analyze finite groups.
Students will use various canonical types of groups and rings.
Students will analyze composition series.

Course Content

Section I: Conjugates and centralizers in Sn, p-groups, Group actions, Counting orbits. Sylow subgroups, Sylow theorems, Applications of Sylow theorems, Description of groups of order p² and pq, Survey of groups up to order 15.
Section II: Normal and subnormal series, Solvable series, Derived series, Solvable groups, Solvability of Sn, Central series, Nilpotent groups, Equivalent conditions for a finite group to be nilpotent, Upper and lower central series. Composition series, Zassenhaus lemma, Jordan-Holder theorem.
Section III: Modules, Cyclic modules, Simple and semi-simple modules, Schur lemma, Free modules, Torsion modules, Torsion-free modules, Torsion part of a module, Modules over principal ideal domain, and its applications to finitely generated abelian groups.
Section IV: Noetherian and Artinian modules, Modules of finite length, Noetherian and Artinian rings, Hilbert basis theorem. Properties of Jacobson radical.

Recommended Books

Luther, I.S., Passi, I.B.S., Algebra, Vol I: Groups, Vol III: Modules, Narosa Publishing House.
Lanski, C., Concepts in Abstract Algebra, American Mathematical Society.
Sahai, V., Bist, V., Algebra, Narosa Publishing House.
Malik, D.S., Mordenson, J.N., and Sen, M.K., Fundamentals of Abstract Algebra, McGraw Hill.
Bhattacharya, P.B., Jain, S.K., and Nagpaul, S.R., Basic Abstract Algebra.
Musili, C., Introduction to Rings and Modules, Narosa.
Jacobson, N., Basic Algebra, Vol I & II, W.H Freeman
Artin, M., Algebra, Prentice-Hall of India.
Macdonald, I. D., The Theory of Groups, Clarendon Press.

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Test your understanding of key group theory concepts including orders of elements, subgroups, and the identity element. This quiz covers various properties and definitions essential in the study of groups and their structures.

Group Theory Concepts Quiz

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Podcast

Questions and Answers

What is the result of the equation $o(x) = p_i m_i$ when $m_i = 0$?

When $x$ is an element of $H_i$, then $o(x)$ is equal to the order of $H_i$.

What does $H_i igcap H_1 H_2 eq ext{e}$ imply?

In the expression $x = (h_1 h_2 ... h_{i-1} h_{i+1} ... h_r)$, $h_t$ is the _____ element.

Match the symbols with their meanings:

Which of the following statements is true regarding the identity element in a group?

In an Abelian group, the operation is commutative.

What is a subgroup of a group G?

A group with infinite elements is called an __________ group.

Match each term with its corresponding description:

Which statement is true regarding the cancellation laws in group theory?

All elements in a group must have unique inverses.

The sets Ha and aH are called __________ of a subgroup H in G.

What is the relationship between the orders of the groups HxK and HxKx−1?

If K is a subgroup of G, then xKx−1 is not a subgroup of G.

What is the highest power of p such that $p^n$ divides the order of G?

According to Lagrange’s Theorem, o(H) o(H ∩ xKx−1) = ______.

Match the following terms with their definitions:

What can be inferred if H and K are two Sylow p-subgroups of G?

If o(H) = o(K) = p^n, then H and K must be equal.

What does the notation $xKx^{-1}$ signify in group theory?

What condition must hold for a subgroup to exist in G when considering the order o(G')?

If pm does not divide the order of any proper subgroup H of G, then pm divides o(G).

What theorem is applied to deduce the existence of an element a in Z(G) such that a^p = e?

The cyclic subgroup generated by an element a is denoted as K = < a > = {a, a 2 , a 3 ,..., a ______}

If $o(G) = pq$, where $p$ and $q$ are distinct primes and $p < q$, what can be deduced about the structure of $G$?

Match the following concepts with their definitions:

The normalizer $N(H igcap K)$ is equal to the group $G$ if and only if $H$ is normal in $G$.

What does the class-equation for G describe?

What is the condition for $N(H igcap K)$ to equal the entire group $G$?

If $o(N(H igcap K)) = 108$, it follows that $G$ is __________.

Every subgroup of Z(G) is a normal subgroup of G.

Match the Sylow theorem notations with their meanings:

In case I, what is assumed about the order of a subgroup H of G?

What condition leads to a contradiction when analyzing the number of Sylow $p$-subgroups?

The Sylow third theorem implies that a group with a unique Sylow subgroup is guaranteed to be abelian.

What can be concluded if $o(H) = p$ and $H$ is unique in $G$?

Which condition must be satisfied for G to be considered an internal direct product of its Sylow subgroups?

Every subgroup of an abelian group is normal in that group.

What is the identity element denoted as in group theory?

If o(H_i) = p_i^{n_i}, then the order of the subgroup H_i is a power of the prime number _______.

Match the concepts with their definitions:

In proving G is an internal direct product, which of the following is not required?

The equation x = h1h2...hi-1hi+1...hr implies that x is an element of Hi for any i.

What is the significance of the notation P_j for j ≠ i in the context of Sylow subgroups?

Study Notes

Course Information

Course Outcomes

Course Content

Recommended Books

Studying That Suits You

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What does $H_i igcap H_1 H_2 eq ext{e}$ imply?

The normalizer $N(H igcap K)$ is equal to the group $G$ if and only if $H$ is normal in $G$.

What is the condition for $N(H igcap K)$ to equal the entire group $G$?

If $o(N(H igcap K)) = 108$, it follows that $G$ is __________.