Prove that the intersection of two subgroups is always a subgroup.

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: is the smallest order of a group that contains both a subgroup isomorphic to Z12 and Z18?

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Q: Suppose that H is a subgroup of Z under addition and that H contains 250 and 350.  What are the…

A: To determine the possible subgroups H satisfying the given conditions

Q: Show that the translations are a normal subgroup of the affine group.

A: To show: The translations are a normal subgroup of the affine group

Q: Assume that (G, ) is a group and that (H, ) and (K, ) are subgroups of (G,*). Prove that (HnK,*) is…

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Q: Show that the center of a group of order 60 cannot have order 4.

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Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

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Q: Show that if H and K are subgroups of a group G, then their intersection H ∩ K is also a subgroup of…

A: Subgroup Test           A subset H C G of the group G will be a subgroup if it satisfies the…

Q: Prove that the intersection of two subgroups of a group G is a subgroup of G.

A: We will prove the statement.

Q: Prove that a group of order 12 must have an element of order 2.

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Q: Prove or give counter example Every characteristic subgroup is fully invariant

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Q: prove that any group R=3 must beperiedio

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Q: Find a proper subgroup of the group of integers ? under addition and prove that this subgroup is…

A: Solution: The objective is to find a proper subgroup of the group of integers and to show that this…

Q: prove that Every group oforder 4

A: Give statement is Every group of order 4 is cyclic.

Q: Prove that a group of order 15 is cyclic

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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: The center of G is a commutative subgroup of G.

A: To prove that the center of G is a commutative subgroup of G.

Q: In Z, find all generators of the subgroup <3>. If a has infinite order,find all generators of…

A: Since a has infinite order, the same holds for a 3 since if it would have order n < ∞, then 1 =…

Q: The intersection of two normal subgroups is always normal sub group. (True/False)

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Q: Show that ( Z,,+,) is a cyclic group generated by 3.

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Q: Prove that S, is isomorphic to a subgroup of An+2-

A: An even permutation can be obtained as the composition of an even number and only an even number of…

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: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Theorem(7.11) : If (H, *) is a subgroup of the group (G, *) , then the pair (NG(H), *) is also a…

A: The normalizer of G, is defined as,  NG(H) = { g in G : g-1Hg = H }

Q: The subgroup {e} is called the nontrivial, that is, a subgroup that is not e is nontrivial.…

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Q: State the first isomorphism theorem for groups and use it to show that the groups/mz and Zm are…

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Q: Every subset of every group is a subgroup under the induced operation. True or False then why

A: True or FalseEvery subset of every group is a subgroup under the induced operation.

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: Is the identity element in a subgroup always going to be the same as the identity of the group?

A: Are the identity elements in a subgroup and the group always the same?

Q: Prove that

A: To prove: Every non-trivial subgroup of a cyclic group has finite index.

Q: Prove that a normal subgroup need not to be a characteristic subgroup.

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Q: Show that every subgroup H of the group G of index two is normal.

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Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

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Q: The identity element in a subgroup H of a group G must be the same as the identity element in G…

A: The identity element in a subgroup H of a group G must be the same as the identity element in G.

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 Z denote the group of integers under addtion. Is every subgroup of Z cyclic? Why? Describe all…

A: Solution

Q: The Kernal of any group homomorphism is normal subgroup True False

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Q: Suppose that X and Y are subgroups of G if |X|=28 and |Y|=42, then what is

A: "According to Bartleby Guideline, Handwritten solution are not provided" Given,            |x|=28…

Q: Prove that a finite group is the union of proper subgroups if andonly if the group is not cyclic

A: union of proper subgroups  proof: Let G be a finite group. In the first place, we are going the…

Q: e subgroups

A: Introduction: A nonempty subset H of a group G is a subgroup of G if and only if H is a group under…

Q: Prove that the intersection of two subgroups of G is itself a subgroup of G.

A: Suppose that, H1 and H2 be the sub groups of G. Also, let H=H1∩H2 . Consider one step group test:…

Q: Suppose that G is a group that has exactly one non-trivial proper subgroup. Prove that G is cyclic.

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Q: Prove that every group of order 78 has a normal subgroup of order 39.

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Q: If H₁ and H₂ be two subgroups of group (G,*), and if H₂ is normal in (G,*) then H₂H₂ is normal in…

A: When a non-empty subset of a group follows all the group axioms under the same binary operation, the…

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: 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: Prove that a normal subgroup must be a union of conjugacy classes.

A: Let N be a normal subgroup of a group G. To exhibit N as a union of conjugacy classes in G.

Q: Show that a group of order 77 is cyclic.

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