Suppose G is a group of order 48, g € G, and g" = €. Prove that g = ɛ.
Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is
A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…
Q: Prove that a group of order 375 has a subgroup of order 15.
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Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…
A: A set G is called a group if it satisfies four properties Closure property: ab∈G where a,b ∈G…
Q: Let (G,*) be an a belian group, if (H,) and (K,*) are subgroup of (G,*) then (H * K,*) is a subgroup…
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Q: Let (G,*) be a group of order p, q, where p, q are primes and p < q. Prove that (a). G has only one…
A: It is given that G, * is a group of order p·q where p, q are primes and p<q. Show that G has only…
Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…
A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…
Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…
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Q: Suppose that G is a group of order 168. If G has more than oneSylow 7-subgroup, exactly how many…
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Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…
A: We need to prove that , group G = a , b with defining sets of relations a3 = e , b7 = e also…
Q: Show that group U(1) is isomorphic to grop SO(2)
A: The solution is given as follows
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: Prove that a group of order 3 must be cyclic.
A: Given the order of the group is 3, we have to prove this is a cyclic group.
Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.
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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: In a group G,let a,b and ab have order 2.show that ab=ba
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Q: et G be a group with order n, with n > 2. Prove that G has an element of prime order.
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Q: Prove that, there is no simple group of order 200.
A: Solution:-
Q: Z, x is not a group.
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Q: Prove that there is no simple group of order 280 = 23 .5 . 7.
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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 (G,*) be an a belian group, if (H,*) and (K,*) are subgroup of (G,*) then (H * K,*) is a…
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Q: Let H and K be subgroups of a group G with operation * . Prove that HK .is closed under the…
A: Given information: H and K be subgroups of a group G with operation * To prove that HK is a closed…
Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…
A: The Sylow theorems are significant in the categorization of finite simple groups and are a key…
Q: If a is an element of order 8 of a group G,
A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e where e is an identity…
Q: Every finite group of order 36 has at most 9 subgroups of order 4 and at most 4 subgroups of order 9…
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Q: If N is a normal subgroup of order 2 of a group G then show that N CZ(G).
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Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.
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Q: = Prove that, there is no simple group of order 200.
A:
Q: Let G be a group of order 24. If H is a subgroup of G, what are all the possible orders of H?
A: Given, o(G)=24 wherre H is a subgroup of G from lagrange's theoram: for any finite order group of G…
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: 7. Prove that if G is a group of order 1045 and H€ Syl19 (G), K € Syl₁1 (G), then KG a and HC Z(G).…
A: As per policy, we are solving only the first Question, Please post multiple Questions separately.
Q: Show that any group of order less than 60 is cyclic
A: This result is not correct. There is a group of order less than 60 which is not cyclic.
Q: For any group G, GIZ(G) is isomorphic to Inn(G)
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Q: If G is a group with identity e and a2 = e for all a ∈ G, then prove that G is abelian.
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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: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…
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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: Prove that any group of order 75 can have at most one subgroup of order 25.
A: Given that a group G with order G=75. We need to prove it has at most one subgroup of order 25. We…
Q: Let G be a finite group of order 125 with the identity element e and assume that G contains an…
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Q: Suppose the o and y are isomorphisms of some group G to the same group. Prove that H = {g E G| $(g)…
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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
A: It is given that, G is a group and g∈G. To sow that if H is a sylow p-subgroup of G, then so is…
Q: 8. Prove that if G is a group of order 60, then either G has 4 elements of order 5, or G has 24…
A: As per the policy, we are allowed to answer only one question at a time. So, I am answering second…
Q: Let G be an abelian group, then (acba)(abc)¯1 is
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Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…
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Q: 7. Prove that if G is a group of order 1045 and H€ Syl₁9 (G), K € Syl (G), then KG and HC Z(G).
A: 7) Let G be a group of order 1045 and H∈Syl19(G) , K∈Syl11(G). To show: K⊲G and H⊆Z(G). As per…
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: : Show that in a group G, if a? = e,Vx E G, then G is a commutative. %3D
A:
Abstract Algebra
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- 15. Prove that if for all in the group , then is abelian.Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.