a. Every finite group is cyclic. b.Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9.

Q: 50

A: From the given information, it is needed to prove or disprove that H is a subgroup of Z:

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: Prove that, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

A:

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:

Q: Label each of the following statements as either true or false   The order of an element of a finite…

A: Theorem :                           In a finite group, the order of a group element…

Q: True or False. Every group of order 159 is cyclic.

A: According to the application of the Sylow theorems, it can be stated that: The group, G is not…

Q: Prove that there is no simple group of order 300 = 22 . 3 . 52.

A:

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: Prove A3 is a cyclic group

A: We know that If G be a group of prime order then G is cyclic group

Q: One of the following is True O If aH = bH, then Ha = Hb. O Only subgroups of finite groups can have…

A:

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: 32. If H and K are subgroups of G, show that Hn K is a subgroup of G. (Can you see that the same…

A: To show:

Q: If H is a Sylow p-subgroup of a group, prove that N(N(H)) = N(H).

A: Let G be a finite group and H be the subset of G. Then, normalizer of H in G, when we conjugate H…

Q: If a is an element of order 8 of a group G, and

A: Let G be a group. Let a is an element of order 8 of group G. That is, a8=e          where e is an…

Q: Let h: G G be a group homomorphism, and gEG is an element of order 35. Then the possible order of…

A:

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: If d divides the order of a cyclic group then this group has a subgroup of order d. Birini seçin: O…

A:

Q: Prove that there is no simple group of order 216 = 23 .33.

A:

Q: It is not possible that, for a group G and H and K are nomal subgroups of G, H is isomorphic to K…

A: Let G be a group and H and K are normal subgroups of G

Q: Prove that there is no simple group of order 280 = 23 .5 . 7.

A:

Q: Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group {e, a, b, a…

A: Given :              A group of order 4  Corollary :                    The order of an element of a…

Q: The following is a Cayley table for a group G. The order of 3 is: 2 4 2 4 2 3 4 1 3 4 5 2 4 ere to…

A: Given Cayley table

Q: Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.

A:

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: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order…

A:

Q: The number of generators of a cyclic group of order 213 is * 48 24 144 140

A:

Q: 5: (A) Prove that, every group of prime order is cyclic.

A:

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: a.) What is the order of any non trivial subgroup of B? b.) What are the trivial subgroups of B?…

A:

Q: ?Which of the following statement is true The largest order among the orders of all the cyclic…

A: We go through each options one by one to choose the correct.

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:…

A:

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: 1. There is no simple group of order 200.

A: Solution:-As per guidelines I submit first question only

Q: 1. State, with reasons, which of the following statements are true and which are false. (a) The…

A: Given Data: (a) The dihedral group D6 has exactly six subgroups of order 2. (b) If F is a free group…

Q: Prove that if G is a finite group, then the index of Z(G) cannot be prime.

A:

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: Prove that every subgroup of nilpotent group is nilpotent

A: Consider the provided question, We know that, prove that every subgroup of nilpotent group is…

Q: Every finite group is cyclic

A:

Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

A:

Q: I'm unsure of how to approach this... Let N be a finite group and let H be a subgroup of N. If |H|…

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

Q: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.

A:

Q: Any group of order 520 is simple

A: Simple group:     "A simple group is a nontrivial group whose only normal subgroups are the trivial…

Q: a) Show that given a finite group G and g ∈ G, the subgroup generated by g is itself a group. (b)…

A:

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-…

A:

Q: If G is a cyclic group, prove for subgroup N that G is a cyclic N

A:

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: Prove that every group of order 375 has a subgroup of order 15.

A: According to the given information it is required to prove that every group of order 375 has 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:…

A:

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…

A: Need to find intersection of subgroup