1. Prove that in any group, an element and its inverse have the same order.

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

A:

Q: If a, b are elements in a group G, show that (ab)-1= b-1a-1

A:

Q: 28. Is every group a cyclic? Why?

A:

Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

A:

Q: 2. Are the groups (R, +) and (R†,') isomorphic? Justify your answer.

A:

Q: True or False. Every group of order 159 is cyclic.

A: According to the application of the Sylow theorems, it can be stated that: The group, G is not…

Q: Prove that there is no simple group of order 300 = 22 . 3 . 52.

A:

Q: (b) Explain how Proposition 3 can be used to show that the multiplicative group Z is not cyclic.

A: The proposition 3 says, if G is a finite cyclic then G contains at most one element of order 2.

Q: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

Q: If a is an element of order 8 of a group G, and

A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e          where e is an…

Q: Verify that (Z, O) is an infinite group, where Z is the set of integers and the binary operator O is…

A:

Q: (d) Define * on Q by a * b = ab. Determine whether the binary operation * gives a group on a given…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: shiw that in a group if x has an inverse y abd a right inverse r, then y and r are the same element

A:

Q: Remark: If (H, ) and (K,) are subgroup of a group (G, ) there fore (HUK, ) need not be a subgroup of…

A: Definition of subgroup: Let (G ,*) be a group and H be a subset of G then H is said to be subgroup…

Q: Suppose that H is a subgroup of a group G and |H| = 10. If abelongs to G and a6 belongs to H, what…

A: Given: H is a subgroup of a group G and |H| = 10 To find: If a belongs to G and a6 belongs to H,…

Q: Prove that there is no simple group of order 216 = 23 .33.

A:

Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

A:

Q: Prove that there is no simple group of order 280 = 23 .5 . 7.

A:

Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove

A:

Q: If G is a group and g E G, show that the number of conjugates of g E G is [G : CG(g)]

A: Given G be a group and g∈G be an element. Let Bg be the set of all conjugate elements of g∈G.…

Q: Prove that in a group, (ab)^2=a^2b^2 if and only if ab=ba.

A: Proof:Let a,b ∈ G.Assume (ab)2 = a2b2 and that prove ab = ba as follows.

Q: If a is an element of order 8 of a group G,

A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e  where e is an identity…

Q: If a1, a2, . . . , an belong to a group, what is the inverse of a1a2 . . . an?

A:

Q: Prove that if (ab)' = a*b² in a group G, then ab = ba.

A: Given,ab2=a2b2To prove: ab=ba

Q: Every commutative group has at least element ??

A: Every commutative group has at least               element ?  We know that , every commutative group…

Q: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

Q: belong to a group. If |a| = 12, |b| = 22, and (a) N (b) # {e}, prove that a® = b'1.

A:

Q: Prove or disprove, as appropriate: If G x H is a cyclic group then G and H are cyclic groups.

A: GIven two groups G and H such that GxH is cyclic. True or false: G and H themselves are cyclic

Q: 1. There is no simple group of order 200.

A: Solution:-As per guidelines I submit first question only

Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

Q: Prove that every subgroup of nilpotent group is nilpotent

A: Consider the provided question, We know that, prove that every subgroup of nilpotent group is…

Q: Every finite group is cyclic

A:

Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: 2. Show that in a group if x has inverse y and y and a right an inverse r, then y and r are the same…

A: We need to show that in group if x has an inverse y and a right inverse r, then y and r the same…

Q: Show that if G and H are isomorphic group, then G commutative implies H is commutative also.

A:

Q: Any group of order 520 is simple

A: Simple group:     "A simple group is a nontrivial group whose only normal subgroups are the trivial…

Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

A:

Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…

A:

Q: Prove that the following set form a group with addition as the operation: {0,5,10,15}

A:

Q: Determine whether the set G is a group under the operation * G={n integer|n is odd}; a*b=a+b

A:

Q: Every element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.

Q: 1,) and (G2,*) be two groups and →G2 be an isomorphism. Then *

A: given that G1,. and G2,*are two groups  and φ:G1→G2  be an isomorphism

Q: G1 = Z÷ and G2 = Z, *

A:

Q: Prove that ifH and K are subgroups of a group G with operation *, Question 8. then HNK is a subgroup…

A:

Q: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

A:

Q: Prove that in any group, an element and its inverse have the same order.

A: Proof:Let x be a element in a group and x−1 be its inverse.Assume o(x) = m and o(x−1) = n.It is…

Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.