If G is a finite group with |G|<180 and G has subgroups of orders 10, 18 and 30 then the order of G is: * 60 90 80 O 30 G1 G2 be an isomorphism. Then

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: 15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk =…

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Q: 3. Let G be a group, H a subgroup of G, and g and element of G. (a) Prove that {ghg-1 | h e H} is…

A: Let G be a group, H is a subgroup of G, and g be an element of G

Q: Suppose that o: G→G is a group homomorphism. Show that () p(e) = ¢(e') (ii) For every gE G, ($(g))-1…

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Q: G. Show that if H and K are subgroups of an abelian group G, then {hk\h E H andke K} is a subgroup…

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Q: Suppose that G is a cyclic group such that Ord(G) = 48. The number of subgroups that G has is * O 8…

A: Q1.  Third option is correct. Q2. Second option is correct.

Q: Let G be a finite cyclic group and a be an element of G of order 8 then the order of G would be: O…

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Q: (G,+) is a finite group such that (a+b)^2 = a^2 + b^2 for all a,b E G. show that G is abelian

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Q: Let H and K be subgroups of a finite group G with H C KC G. Prove that |G:HI |G:K| |K:H].

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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: Prove that if H is cyclic, then ø[H] is also cyclic.

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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: Suppose H and K be subgroups of a finite group G with |G : H| = m and |G : K| = n. Prove that…

A: We use here, Tower law of subgroup which states that Let (G,∘) be a group.  Let H be a subgroup of G…

Q: Let G be an abelian group and suppose that H and K are subgroups of G. Show that the following set…

A: According to the given information, Let G be an abelian group and suppose that H and K are subgroups…

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: Let G be a group with |G|=187 then every proper subgroup of G is:

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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: 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 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: 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: {hk | h ∈ H, k ∈ K}}

A: We have to prove that  {hk|h∈H, k∈K} is a subgroup of G.

Q: Prove property 1

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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: Let G be a finite group and H1, H2, ., Hk be subgroups of G. (a) Show that k H1n H2n. n Hk < G. Hi…

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Q: 8. Let (G,*) be a group, and let H, K be subgroups of G. Define H*K={h*k: he H, ke K}. Show that H*…

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Q: 5. Let (G, *) be a cyclic group, namely G=(a). Prove that (a). if G is finite of order n, then G…

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Q: 5.22 Let G be a group. Prove that Z(G) is a subgroup of G.

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

A: If G is cyclic group and order of G is 'n'. Then number of subgroups of G is equal to number of…

Q: Let G be a finite cyclic group and a be an element of G of order 9 then the order of G would be: *…

A: Let G be a finite group and 'a' be an element  in G  s.t O(a)=9

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: 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…

A: It is given that, H is a subgroup of G and |x|=n. Also  k is the least positive integer such that…

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

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Q: If G is a finite group with IG|<160 and G has subgroups of orders 10, 16 and 20 then the order of G…

A: Given :- order of a group G... |G| &lt; 160 Also order of subgroups of group G are 10, 16 , 20. We…

Q: Let G be a finite cyclic group and a be an element of G of order 8 then the order of G would be: 2…

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Q: 4. Let G be a group and let H, K be subgroups of G such that |H| = 12 and |K| = 5. Prove that HNK =…

A: We have to prove given result:

Q: If G is a finite group with IG|<120 and G has subgroups of orders 10, 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 A is a subgroup of G,…

Q: Let G be a finite cyclic group and a be an element of G of order 9 then the order of G would be: * 4…

A: Fourth option is correct.

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 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 K and H be subgroups of a finite group G with KCHCG.lf [G:K] = 12 and [H:K] = 3. Then, [G:H] =…

A: Let , K  and  H  be subgroups of finite group G. Also . K ⊆ H ⊆ G Here , G : K = 12  , H : K = 3 We…

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: 3. Let G be a finite group with an odd number of elements. Prove that G is isomorphic to a subgroup…

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Q: 2. Let G be a group of order /G| = 49. Explain why every proper subgroup of G is сyclic.

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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: 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: Show that the following two conditions on a group G are equivalent: homomorphism p from G into Σ,…

A: We have to prove that the following two conditions on a group G are equivalent : (1) There is a…

Q: If G is a finite group and some element of G has order equal to the size of G, we can say that G is:…

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