If G is a finite group with |G|<120 and G has subgroups of orders 1O, 15 and 20 then the order of G is: * O 60 О 30 O 90 O 80 ООО О

Q: Let G be a finite group of order n. Let g be an element of G. Prove that gn=e.

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Q: . Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the…

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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: Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G. Prove that if…

A: Given that, Let G be a finite group, let H be a subgroup of G and let N be a normal subgroup of G.

Q: Let G be a finite cyclic group of order 20, and a in G. Then one of the following is possible order…

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Q: Let a be an element of a group G such that Ord(a) = 30. If H is a normal subgroup of G, then Ord(aH)…

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

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Q: Let H be a subgroup of a group G and a, be G. Then be aH if and only if None of these O ab e H O…

A: We know that b∈bH                                    (1) We know that aH = bH if and only if a-1b∈H…

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 N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the…

A: According to the given information, Let N be a finite group and H be a subgroup of N. H is odd and…

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: 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 with |G|=187 then every proper subgroup of G is: * O Non abelian O Non cyclic None…

A: Given: G is a group with order 187. Theorems used: Any group of prime order is cyclic Lagrange…

Q: Let G be a group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

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

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Q: 5. Suppose G is a group of order 8. Prove that G must have a subgroup of order 2.

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Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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

A: an abelian group, also called a commutative group, is a group in which the result of applying the…

Q: Let K and H be subgroups of a finite group G with KCHCG. If [G:H] = 4 and [H:K] = 3. Then, [G:K] =…

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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 with |G|=221 then every proper subgroup of G is: O Cyclic O None of these O Non…

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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: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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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: 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 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: 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 with p or more distinct elements of order p for some prime p. Show that G is not…

A: Given : Let G be a group which has p  elements of order p, p is a prime no.  To show G is not cyclic…

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…

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: 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 G be a group with |G|=187 then every proper subgroup of G is: * O Non cyclic O Non abelian O…

A: In the given question we have to choose the correct option from the given options. In the solution…

Q: Let H be a subgroup of a group G and a, be G. Then b E aH if and only if ab-1 e H O ab e H O None of…

A: Ans is given below

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 a be an element of a group G such that Ord(a) = 32. If H is a normal subgroup of G, then Ord(aH)…

A: Result: Let G be a group and H be a normal subgroup of G. Let 'a' be an element of G such that order…

Q: Prove that if G is an abelian group of order n and s is an integer that divides n, then G has a…

A: G is an abelian group of order n ; And, s is an integer that divides n;

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: Let G be a group with G|=221 then every proper subgroup of G is: " O Non cyclic O None of these 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: If G is a finite group with |G|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

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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