Construct the distinct left cosets of the subgroup {(1), (1,3,2), (1,2,3)} in S3.

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A: We have to choose the correct option.

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A:

Q: 7. Show that 4 is a subgroup of S,

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A: Given that 2Z is the set of all even integers.

Q: Find all the conjugate subgroups of S3, which are conjugate to C2 .

A: Given-S3 To find- all the conjugate subgroup of S3 which are conjugate

Q: Which among is a non-cyclic group whose all proper subgroups are cyclic? U(12), Z8 , Z, U(10)?

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: Given: 2Z is a subgroup of (Z,+). We have to find the right coset of -5+2Z.

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: 6. List every generator for the subgroup of order 8 in Z32.

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A:

Q: (c) Prove that the intersection of any three subgroups is a subgroup while the union of two…

A:

Q: Find the cyclic subgroup of S6 generated by the element (123)(456)

A: If G is a group and g is an element of G of order n, then the cyclic subgroup of G generated by g is…

Q: Q/ In (Z6 , +6 ) find the cyclic subgroup generated by 1, 2, 5.

A:

Q: Which of the following cannot be an order of a subgroup of Z12? 12, 3, 0, 4?

A: Since 0 does not divides 12.

Q: Show that S5 does not contain a subgroup of order 40 or 30.

A: Let’s assume that the H is a subgroup of S5. So,

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A:

Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

A:

Q: The center of G is a commutative subgroup of G.

A: To prove that the center of G is a commutative subgroup of G.

Q: In Z, find all generators of the subgroup <3>. If a has infinite order,find all generators of…

A: Since a has infinite order, the same holds for a 3 since if it would have order n < ∞, then 1 =…

Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is

A: It s given that H and K are subgroups of G, H=18 and K=30. Since H, K are subgroups, H∩K≤H  and…

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A:

Q: Which of the following cannot be an order of a subgroup of Z12? 4 3 Option 4 12

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -3 + 2Z contains the…

A: Given: 2Z is a subgroup of (Z,+). We have to find the right coset of -3+2Z.

Q: In Z24, list all generators for the subgroup of order 8. Let G = <a>and let |a| = 24. List all…

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Q: Find the three Sylow 2-subgroups of S4

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Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

Q: Show that the subgroup generated by any two distinct elements of order 2 in S3 is all of S3.

A: Given that, S3 is a symmetric group of permutations. Thus, S3 has 6 elements. By using Lagrange's…

Q: 3) Show that the subgroup of Dg is isomorphic to V4.

A:

Q: Show that in C* the subgroup generated by i is isomorphic to Z4.

A: C* is group of non-zero comples numbers with multiplication

Q: Find four different subgroups of S4 that are isomorphic to S3.

A: If there exists an isomorphism between the subgroups they are called isomorphic ( Isomorphism is…

Q: At now how many elements can be contained in a cyclic subgroup of ?A

A: There will be exactly 9 elements in a cyclic subgroup of order 9.

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: Suppose G is a group and H is a subgroup of G .Then set a+H is left coset if H in G . set H+a is…

Q: One of the following is a p-sylow subgroup of the group Z, × Zg H={(0,0),(0,4)} O…

A:

Q: List all of the elements in each of the following subgroups. (4) The subgroup of GL2(R) generated…

A: (4) Let A=1-11  0 Then,  A2=A·A   =1-11  0·1-11  0   =0-11  -1 A3=A·A2   =1-11  0·0-11  -1   =-1  00…

Q: The set A := {(1), (1 2 3), (2 3 4)} forms a subgroup of the permutation group (G,.). O True O False

A:

Q: Show that every subgroup H of the group G of index two is normal.

A:

Q: Let (Z12, +12) be a group , if we take {0,4,8} for the set H then ({0,4,6}, +12) is evidently a…

A: Let H=0, 4, 6 We know that the operation in ℤ12 is addition. So, the element of left coset is of the…

Q: (Z., +6) is a subgroup of (Zg, +8)

A:

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -5 + 2Z contains the…

A: 7 is the correct answer.

Q: Find all cosets of the subgroup of Z12.

A:

Q: Q2/ In (Z9, +9) find the cyclic subgroup generated by 1,2,5

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Q: Find a subgroup of Z12 ⨁ Z4 ⨁ Z15 that has order 9.

A: Given group is Z12⊕Z4⊕Z15. It is known that for each divisors r of n, Zn has exactly one cyclic…

Q: Consider the "clock arithmetic" group (Z12,O), together with subgroup H= {0,4, 8} Write all cosets…

A: Consider the ‘clock arithmetic’ group Z12, ⊕, together with subgroup H=0,4,8. The objective is to…

Q: The set of all even integers 2Z is a subgroup of (Z, +) Then the right coset -6 + 2Z contains the…

A: 10 is the element in the right coset.

Q: Find all the Sylow 3-subgroups of S4.

A:

Q: Prove that a group of order 595 has a normal Sylow 17-subgroup.

A:

Q: b. Find all the cyclic subgroups of the group ( Z6, +6).

A:

Q: b. Find the center and the commutator subgroup of S2 x Z7.

A: Now we knew that  Z2 is isomorphic to S2. So it is commutative group. The center subgroup of G := S2…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.