Let G be a finite group of order n and identity element 1. a. Show that g" =1 for any g e G. b. An element g € G is said to be square if there exists z e G such that g = x?. Prove that G has odd order if and only if every element of g is a square.

Q: Let G be a group of order pm where p is a prime number and m is a positive integer. Show that G…

A: Let G be a group of order pm, where p is a prime number and m is a positive integer. Then,…

Q: Let G be the subgroup of GL3(Z₂) defined by the set 100 a 10 bc1 such that a, b, c Z₂. Show that G…

A: The given set of matrix is 100a10bc1 where a, b, c∈ℤ2. To find: the group to which the given set is…

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Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

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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 H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

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: Let G be a finite group and H1, H2,…., Hk be subgroups of G. .... (a) Show that N H; = Hị n H2 n..n…

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Q: Let G be a group of order 100 that has a subgroup H of order 25.Prove that every element of G of…

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Q: Let G be a finite group with no element of order 3, and let a E G. (a) Can 3 divide o(a)? Either…

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Q: Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

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Q: Suppose S is a nonempty subset of a group G.(a) Prove that if S is finite and closed under the…

A: (a)Suppose S is a non-empty subset of a group G. then we have to prove that if S is finite and…

Q: If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G…

A: Using Lagrange's theorem it can be written that if G is a finite group and B is a subgroup of G,…

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 finite group of order p  where p is prime. Prove that G is cyclic.

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Q: 3.) Prove that for every element a of a Group G, Z(G) is a subset of C(a)

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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 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 group and let H and K subgroup of G. Prove that the intersection H and K is a subgroup of…

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Q: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

A: Given orders of subgroup 10 18 30

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: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

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Q: (a) Prove that every element of Q/Z has finite order. (b) Given two groups (G, .) and (H, *).…

A: This question is related to group theory.

Q: Let G be a finite group and let sym(G) be the group of all permutations on G. For each g in G, let…

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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: If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G…

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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: Let G be a group and H ≤G. Let x ∈G be an element of finite order n. Prove that if k is the least…

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Q: Let G be a group. Prove that Z(G) is a subgroup of G.

A: The set ZG=x∈G|xg=gx,∀g∈G of all elements that commute with every other element of G is called 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: If op is a homomorphism of group G onto G with kernel K and Ñ is a normal subgroup of G. N = {x E G|…

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Q: 2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove…

A: Given   ϕ:G -&gt; G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: 7. Let G be a group, and let g E G. Define the centralizer, Z(g), of g in G to be the subset Z(g) =…

A: let G be a group, and let g∈G. Define the centralizer, Zg of g in G to bethe subset…

Q: Let G be a group of order p™ where p is a prime number and m is a positive integer. Show that G…

A: given G be a group of order pm where p is a prime number and m is a positive integer to show G…

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

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Q: If G is a finite group with |G|<180 andG has subgroups of orders 10, 18 and 30 then the order of G…

A: Use the fact that order of subgroup divides order of group

Q: Let G be the additive group R × R and H = {(x, x) : x E R} be a subgroup of G. Give a geometric…

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Q: Let G be a finite group and let H be a normal subgroup of G. Provethat the order of the element gH…

A: Given: G be a finite group and H be a normal subgroup of G.

Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: Let G be a group with order n, with n> 2. Prove that G has an element of prime order.

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

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Q: Prove that if H is normal in G and ø is onto, then ø[H] is normal in G'.

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Q: Let G be a group with no proper, nontrivial subgroups and assume that |G| > 1. Prove that G must be…

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Q: 6. Let G be a group of order p², where p is a prime. Show that G must have a subgroup of order p.

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Q: Let G be a group of order 24. Suppose that G has precisely one subgroup of order 3, and one subgroup…

A: Theorem : If a group G is the internal direct product of subgroups H and K, then G is isomorphic to…