by LetG = {(ª : a, b, , c, d e Z under addition let H EG : a +b + c + d = 1 € Z} H is a %3D subgroup of G.

Q: Enter the smallest subgroup of M2(ℝ)× containing the matrix (−2 −1…

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Q: - Show that the following subset is a subgroup. H = {o e S, l0(n) = n} S,

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Q: How many subgroups of are equal to the subgroup ?

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Q: In (Z13, +13) the set K = {0,3,5,8,9,11} is eyclie subgroup. True False

A: Use the definition and properties of a cyclic group. Check whether the whole group can be generated…

Q: If H and K are subgroups of G, |H|- 16 and K-28 then a possible value of HNK| is 16

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Q: Determine which of the following is normal subgroup O S3 O None of them O SL(2, R) O GL(2,R)

A: We will prove that SL(2,R) is normal subgroup of GL(2,R)

Q: (3) Let (A, +..) be a subgroup of (M₂ (Z), +,.), Then A is ideal of M₂ (Z), where A = {(a b) la, b,…

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Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.

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Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: When we say xH = Hx where H is a normal subgroup of G and x is an element of G, what exactly does…

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Q: Show that no proper subgroup of S4 contains both (1,2,3, 4) and (1,2).

A: We have to show that no proper subgroup of S4 contains both 1,2,3,4 and 1,2. We know that the…

Q: Let n E Z be an integer number and let H = (n) = {rn,r E Z} i.e\ H = (n) = {.., -3n, –2n, -n, 0, n,…

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Q: is a subgroup of Z1, of order: 3 12 O 1 The following is a Cayley table for a group G. 2. 3.4 = 2 3…

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Q: Is the set {3m + v3ni|m, n E Z, b|m – n} the normal subgroup of the (C, +)group?

A: given :

Q: Determine which of the following is a normal subgroup SL(2, R) Z, None of them S3 GL(2, R)

A: Zn is not a sub-group but the subgroups of Zn are normal subgroups.

Q: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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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: O Find Subgroup of order a, 3,4 and 6

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Q: If H and K are subgroups of G, IH|= 16 and |K|=28 then a possible value of IHNK| is 16 8. Activate…

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Q: If H and K are subgroups of G, IH|= 20 and |K|=32 then a possible value of |HNK| is O 2 O 8 O 16

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Q: Consider find Subgraup. Dihedral group D- of order 2,3,4 and 6.

A: A group G of two generators x and y of order n and 2 respectively with some relation is called the…

Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).

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Q: Let G be a group, H4G, and K < G. Prove that HK is a subgroup of G. Bonus: If in addition K 4G,…

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Q: is a subgroup of Z, of order 12 O 12 5, 3.

A: The explanation of the solution as follows:

Q: If H and K are subgroups of G, |H|= 18 and |Kl=30 then a possible value of |HNK| is O18 8. O 4

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Q: G = (R, +), H = {a+bv2: a,b € Z}

A: Given G = (R, +), H = {a+b√2 : a, b∈Z}. We check whether H is a subgroup of G.

Q: If H and K are normal subgroups of G, show that their intersection is also a normal subgroup. To do…

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Q: If H and K are subgroups of G, IH|= 16 and |KI=28 thena possible value of |HNK| is 8. 6. 16

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Q: 10) Which of the following is not a group? * (Z,*), a* b = a + 2b Va, b E Z O (Z,*), a* b = a +b Va,…

A: Option (1) is correct.

Q: let H be a normal subgroup of G and let a belong to G. if th element aH has order 3 in the group G/H…

A: H is normal subgroup of G. And a belongs to G.  O( aH) = 3 in G/H and O(aH) in G/H divides  O(a) in…

Q: 3. How many cyclic subgroups does S3 have?

A: The objective is to find the number of cyclic subgroups of S3. Subgroups of S3 are, H1=IH2=I, 1…

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

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

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Q: Review the normal subgroups N = {(1), (1 2 3), (1 3 2)} of S3. From these subgroups and groups,…

A: This is a problem of Group Theory.

Q: Let G = Z, be the cyclic group of order n, and let S c Z, \ {0}, such that S = -S, \S| = 3 and (S) =…

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Q: Suppose the Cayley table for B = {e, a, b, c}unde the binary operation * is given by * e a e e a a a…

A: Given  B = {e,a,b,c} and the Cayley table  * e a b c e e a b c a a e c b b b c e a c c b…

Q: a b с d e f a b c d e f cdf a b e d e f a ecdb a b fc a b c d e f b f Answer: deca ecbfad Enter a…

A: We have to solve given problem:

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: Let G Są and let K = {1,(1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. K is a normal subgroup of G. What is…

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

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Q: How many subgroups does Z/60Z have?

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Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is

A: It is given that H and K are subgroups of G and H=16, K=28. Since H and K are subgroups of G, H∩K≤H…

Q: Answer the followings: 1. Let H = {[a b]: a, b, d € R, ad # 0}. Is H a normal subgroup of GL₂(R)? 2.…

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Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

Q: If H and K are subgroups of G of order 75 and 242 respectively, what can you say about H N K?

A: Solution

Q: Determine which of the following is a normal subgroup O GL(2. R) SL(2. R) O None of them Os. S,

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Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

Q: Suppose that N and M are two normal subgroups of a group and that IOM = {e}. Show that for any n E…

A: Given: N and M are two normal subgroups of G and N ∩ M = {e} To prove: nm = mn for any n∈ N and m∈ M

Q: Let S be a subset of Sn where S = {a € Snla(1) = 1}. Show that S is a subgroup of Sn.

A: In order to prove S is a subgroup of Sn, prove that α∘β∈S such that α,β∈Sn and α-1 ∈S .

Q: A Let (A, +,.) be a subgroup of (M₂ (Z), +,.), Then A is ideal of M₂ (Z), where = {(ab) a, b, c €…

A: Let I be a non-empty subset of a ring (R,+,·). Then I is an ideal of (R,+,·) if and only if the…

Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent. 

Tree diagram

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described. 

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.