Show that if G is a finite group of even order, then there is an a EG such that a is not the identity and a? = e. Let G be a group and suppose that. (ab)² = a²b² for all a and b in G. Prove that G is

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…

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Q: Prove: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a =…

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Q: Let x be an element of group G. Prove that if abs(x) = n for some positive integer n, then x-1 =…

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Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, bEG is either a left or a right coset…

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Q: Let H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

Q: Show that if G is a cyclic group then G is abelian. : Let n>0 be a positive integer, and let a and b…

A: Let G be a group and let ZG be the center of G. Here, ZG = a∈G / ag = ga , ∀ g ∈ G We need to show…

Q: : Show that if G is a cyclic group then G is abelian. : Let n>0 be a positive integer, and let a and…

A: We will use basic knowledge of group theory to answer this. You have explicitly written on the top…

Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

Q: If G is a finite group with |G|<160 and G has subgroups of orders 1O, 16 and 20 then the order of G…

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Q: Let G be any group and p: G -G, (g) =g1 Vg eG. Then: (a) o is a homomorphism (b) o is a homomorphism…

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Q: Let G be a finite group of order n, positive integer, with identity e. Prove or disprove: For any g…

A: Let G be a finite group of order n, with identity e i.e, order of (G) = |G| = n .   Now let g be any…

Q: Suppose G is a finite group of order n and m is relatively prime to n. If g EG and g™ = e, prove…

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Q: Let G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

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Q: Prove that if G is a finite cyclic group, Hand K are subgroups of G, and H =f. K, then IHI =f. IKI

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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: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

Q: Let G be a group and a e G. Show that o(a) = o(a-). order n, then ba also has order n.

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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: Let l be an integer greater than 1, and let G be a finite group with no element of order l. Can…

A: Find the attachment for the solution.

Q: Let G be a finite group. Let x E G and let G be the conjugacy class of r. Prove that |#C| < |[G,…

A: Given G be a finite group. Let x∈G. Define: xG=gxg-1: g∈G Let G:G=m. Define a map π:G→GG:G defined…

Q: If G is a finite group with |Gl<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

A: Given: The group G is a finite group with | G | &lt; 120 and G has subgroups of orders 10, 15, and…

Q: Let G be a finite group. Let E G and let xG be the conjugacy class of x. Prove that x| < |[G, G]],…

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Q: Prove that the trivial representation of a nite group G is faithful if and only if G = {1G}

A: To Prove: The trivial representation of the nite group G is faithful if and only if G =1G

Q: : Show that if G is a cyclic group then G is abelian. : Let n>0 be a positive integer, and let a and…

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Q: Let G be any group with the identity element e. With using the Group Homomorphism Fundamental…

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Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

Q: If G is a group and g E G, the centralizer of g E G, is the set CG(g) := {a E G : ag =ga} that is,…

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Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 ano then the order of G is:…

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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: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

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Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

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Q: Prove that if G is a finite group and a ∈ G, then the order of a divides the order of G.

A: Given that G is a finite group and a∈G. To prove the order of a divides the order of G: Let the…

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 G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

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Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…

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Q: Let G be a group and a e G such that o(a) = n < oo. Show that a = a' if and only if k =l mod n. %3D

A: Let G be a group and a∈G such that Oa=n&lt;∞. Show that ak=al if and only if k≡l mod n. If k=l the…

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

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Q: If H is a subgroup of a group G such that (aH)(Hb) for any a, b eG is either a left or a right coset…

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Q: A subset of H of a group G is a subgroup of G if the operation on G makes H into a group. Prove that…

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Q: Let G be a group and g E G. Prove that if H is a Sylow p-group of G, then so is gHg-1

A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…

Q: Suppose that G is a cyclic group such that Ord(G) = 54. The number of subgroups that G has is * 10 O…

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Q: (a) If G is abelian and A and B are subgroups of G, prove that AB is a subgroup of G. (b) Give an…

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Q: Let H and K be subgroups of a finite group G. Show that |HK |HK= |HОКI where HK (hk hE H, k E K}.…

A: let D = H ∩K then D is a subgroup of k and there exist a decomposition of k into disjoint right…

Q: Let G be a group with identity e and a € G. (a) Define |G|, the order of G, and |al, the order of a.…

A: (a) The cardinality or the number of elements in a group is called the order of that group. If group…