If a is an element of order 8 of a group G, and
Q: 1. Let a and b be elements of a group G. Prove that if a E, then C. 2. Let a and b be elements of a…
A:
Q: Prove that any group with prime order is cyclic.
A: Given, Any group with prime order. let o(G)=p (p is a prime number) we assure that G has no subgroup…
Q: 28. Is every group a cyclic? Why?
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Q: 1. Prove that in any group, an element and its inverse have the same order.
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Q: Are groups Z×10and Z×12 isomorphic
A: Concept:
Q: Let G a finite group, g E G such
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Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * True False
A: let H be a cyclic subgroup of a group G.
Q: Give an example of a p-group of order 9.
A: Given, Give an example of a p-group of order 9.
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 = ,then one of the following is a possible value of…
A: Given that a is an element of order 8 and a4=ak
Q: (a) What does it mean for two groups to be isomorphic?
A: see my solution below
Q: If G is a finite group, H ≤ G, the order of H divides the order of G: | H | / | G | Prove
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Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.
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Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…
A: The Sylow theorems are significant in the categorization of finite simple groups and are a key…
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: In group theory (abstract algebra), is there a special name given either to the group, or the…
A: Yes, there is a special name given either to the group, or the elements themselves, if x2=e for all…
Q: 27. If g and h have orders 15 and 16 respectively in a group G, what is the order of (9) n (h)?
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Q: Prove that in a group, (a-1)-1 = a for all a.
A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…
Q: If N is a subgroup of an Abelian group, prove that is Abelian. N |
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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: An element a of a group G has order n E z+ if and only if a" = e.
A: The given statement is False.
Q: order 8 of a group G, and =
A: Given that order of a is 8 .Then a8=e Rearrange a little bit , we can have a42=e Hence order of…
Q: Prove that in a group, (a-1)¯' = a for all a.
A: To prove that in a group (a-1 )-1=a for all a.
Q: Show that the multiplicative group Zfi is isomorphic to the additive group Z10.
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Q: Suppose x is an element of a cyclic group of order 15 and x3 = x7 = x°. Determine |x13].
A: According to a theorem in group theory , If G is a finite group and a∈G be an element in the group…
Q: 1. There is no simple group of order 200.
A: Solution:-As per guidelines I submit first question only
Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')
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Q: 1. Let G be a cyclic group of order 6. How many of its elements generate G?
A: Any finite cyclic group of order 'n' has total ϕ(n) number of generators. where 'ϕ' represents…
Q: If a is an element of order 8 of a group G, and 4 = ,then one of the following is a possible value…
A:
Q: 3. Prove or disprove: If G is a group, (g¬')-' = g.
A: Consider the given information: Let G is a group. To show that (g-1)-1=g
Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…
A: The identity element in a subgroup H of a group G must be the same as the identity element in G.
Q: Show that if G is a finite group with identity e and with an even number of elements, then there is…
A: Given,If G is a finite group with identity e and with an even number of elements,then there is a ≠e…
Q: Let x be in a group G. If x' - e and x* - e , prove that x - e and x' = e
A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…
Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).
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Q: If H is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…
A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.
Q: Let G be an abelian group, then (acba)(abc)¯1 is
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Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…
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Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N
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Q: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False
A: we will give the counter example in support of our answer.
Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…
A: An automorphism of a group is the permutation of the group which preserves the property ϕgh=ϕgϕh…
Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…
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Q: Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the…
A: We know that, Order of the given group is divisible by natural numbers 5,7,8 and 9. So the least…
Q: Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.
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Q: : Show that in a group G, if a? = e,Vx E G, then G is a commutative. %3D
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Q: Let G be a finite non-abelian simple group and let q be prime, then [G] is
A: It is given that G be any finite non Abelian simple group and q be any prime. We have to determine…
Q: If a group G is isomorphic to H, prove that Aut(G) is isomorphic toAut(H)
A: We have to prove, If a group is isomorphic to H, then Aut(G) is isomorphic to Aut(H).
Q: Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element.…
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- Let n be appositive integer, n1. Prove by induction that the set of transpositions (1,2),(1,3),...,(1,n) generates the entire group Sn.If a is an element of order m in a group G and ak=e, prove that m divides k.42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .
- Construct a multiplication table for the group G of rigid motions of a rectangle with vertices 1,2,3,4 if the rectangle is not a square.12. Find all homomorphic images of each group in Exercise of Section. 18. Let be the group of units as described in Exercise. For each value of, write out the elements of and construct a multiplication table for . a. b. c. d.Exercises 19. Find cyclic subgroups of that have three different orders.