Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itselt is
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 a finite group , IGI=ps. p prime Prove that G cannot have two distinct and sep. subgroups…
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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: ) If G is a group and X is a G-set, then the subset (g in G: for all x
A: To explain the given statement is true or false as,
Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…
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Q: c) Show that if G is a group of order 100, then G has at most one subgroup of order 25.
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Q: 5. Let p and q be two prime numbers, and let G be a group of order pq. Show that every proper…
A: We have to prove that: Every proper subgroup of G is cyclic. Where order of G is pq and p , q are…
Q: Give the subgroup diagram for each of the groups: (a) Z24 (b) Z36-
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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: If H is a normal subgroup of a finite group G and |H| = pk for someprime p, show that H is contained…
A: H is a normal subgroup of a finite group G and |H| = pk for some prime p.
Q: Let G be a group, and let X be a set. Let I be the intersection of all subgroups of G that contain…
A: Let G be a group and X be a set in G. Suppose I is the intersection of all subgroups of G that…
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: As you asked multiple questions , I answered only first question. Here we can say from a corollary…
Q: Let G be a finite group. Then G is a p-group if and only if |G| is a power of p. We leouo the
A: Given G is finite group and we have to prove G is a p-Group of and only if |G| is a power of p.
Q: Let Z denote the group of integers under addition. Is every subgroup of Z cyclic? Why? Describe all…
A: Yes , every subgroup of z is cyclic
Q: Let G be a group with |G|=187 then every proper subgroup of G is:
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Q: Let (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…
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Q: List all of the subgroups of the group (Z20. +). empty subset of G tha
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Q: If N is a normal subgroup of a group G, and if every member of N and G/N have a finite order, prove…
A: Given: If N is a normal subgroup of a group G, and if every member of N and GN have a finite order…
Q: prove That :- let H and K be subgroups of agroupG of the m is ormal a HK is subgroup of G →1f one
A: Subgroup of a group G
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: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…
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Q: If H is a cyclic subgroup of a group G then G is necessarily cyclic * O True False
A: this is false because this is need not be true because Z4×Z6 Is not cyclic but have
Q: Give an example of subgroups H and K of a group G such that HKis not a subgroup of G.
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Q: Let G be a group of order 90. show that G has at most one subgroup of order 45
A: Given: G be a group of order 90
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A: Here given G is simple group and K is a normal subgroup of G. Then use the definition of simple…
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: Prove that if G is a finite group and H is a proper normal subgroupof largest order, then G/H is…
A: Given: G is a finite group and H is a proper normal subgroup of largest order.
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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…
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Q: Prove that a group that has more than one subgroup of order 5 musthave order at least 25.
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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: 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: If H is the subgroup of group G where G is the additive group of integers and H = {6x | x is the…
A: Let H is a subgroup of order 6 . Take H=6Z where Z is integers.
Q: G is a finite group of order IGI =pqr with p< q <r prime
A: Solution
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Q: If a subgroup H of a group G is cyclic, then G must be cyclic. Select one: O True O False
A: we will give the counter example in support of our answer.
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: 2) Let (G, *) be a group and H, K be subgroups in G. Prove that subset H * K is a subgroup if and…
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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: 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: i have included a picture of the question i need help understanding.thank you in advance. please…
A: Let H and K are two subgroups of the group G.To show
Q: Suppose that G is a group such that Ord(G) = 36. The number of subgroups that G has is 4 O 12 O 18…
A: Given order of G is 36 So U(G) = {1,5,7,11,13,17,19,23,25,29,31,35} So number of elements are 12…
Q: Prove that every group of order 375 has a subgroup of order 15.
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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: Suppose that G is a group such that Ord(G) = 36. The number of subgroups %3D that G has is 4 О 12 О…
A: Order of a group: Let G be a group and n be the number of elements in the group. Then, order of…
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- Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?Let G be a group and gG. Prove that if H is a Sylow p-group of G, then so is gHg1