Verify the corollary to the Fundamental Theorem of FiniteAbelian Groups in the case that the group has order 1080 and thedivisor is 180.

Q: Suppose G is a cyclic group with an element with infinite order. How many elements of G have finite…

A: Suppose G is a cyclic group with an element with infinite order.  It means that order of group is…

Q: Write the Caley table of all abelian groups of order 8

A: We have to write cayley table of abelian group of order 8.

Q: Write under isomorphism all the abelian groups of order 8 and their respective subgroups.

A: To find all the abelian groups of order 8, we can use the fundamental theorem of finitely generated…

Q: Prove that the subset of elements of finite order in an Abelian groupforms a subgroup. (This…

A: To Prove: The subset of elements of finite order in an abelian group forms a subgroup.

Q: Find all abelian groups (upto isomorphism) of order 3375.

A: We have to find the all abelian groups (upto isomorphism) of order 3375.

Q: Show that the group of automorphism of a cyclic group of degree four is of order two.

A: Claim :  The group of automorphism of a cyclic group of degree four is of order two.

Q: An element is called a square if it can be expressed in the form b2for some b. Suppose that G is an…

A: Given: An element is called a square if it can be expressed in the form b2 for some b.G is an…

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

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Q: Suppose that the order of some finite Abelian group is divisible by10. Prove that the group has a…

A: Given: The order of some finite Abelian group is divisible by 10.We need to prove that the group has…

Q: Find any case in which the number of subgroups with an order of 3 can be exactly 4 in the Abelian…

A: Let G be an abelian group of order 108 Find the number of subgroups of order 3. Prove that, in any…

Q: Show that the order of the group of  automorphism of a finite group of  order n is a divisor of n !.

A: we have to show that the order of the group of automorphism of a finite group of order n is a…

Q: How many nonisomorphic abelian groups of order 80000 are there?

A:

Q: Determine the group Z42/ and find a group that it is isomorphic to it. Generalize your observation.

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Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

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Q: Show that the order of the group of automorphism of a finite group of order n is a divisor of n!.

A: we have to show that the order of the group of automorphism of a finite group of order n is a…

Q: Show that the integers have infinite index in the additive group of rational numbers.

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Q: Let G be a group (not ncesssarily an Abelian group) of order 425. Prove that G must have an element…

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Q: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

Q: Find all abelian groups (upto isomorphism)  of order 67

A: We have to find all abelian groups (upto isomorphism) of order 675.

Q: Lagrange's theorem, in group theory, a part of mathematics, states that for any finite group G, the…

A:

Q: Without using the Generalized Cayley Theorem or its corollaries,prove that there is no simple group…

A: A simple group is a group with no non-trivial proper normal subgroups. We need to show that there is…

Q: (6) Let G be a noncommutative group. Show that G has a nontrivial subgroup.

A: We recal Cauchy's theorem and then use it

Q: Show that the order of each element of the quotient group Q/Z is finite.

A: We have to show that the order of each element of the quotient group Q/Z is finite.

Q: If the generator of a cyclic group G is of infinite order (or of zero order), then Gis isomor phic…

A: The given statement is:If the generator of a cyclic group is of infinite order, then is isomorphic…

Q: The order of identity element equal itself in any group

A: .

Q: Show that the multiplicative group Zfi is isomorphic to the additive group Z10.

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Q: 2. What is the cyclic subgroup of Z generated by -1 under +?

A: Let, H = <-1> Now, for any positive integer k,       -k = (-1) + (-1) + (-1) + . . . . . . . .…

Q: Show that any finite subgroup of the multiplicative group of a fieldis cyclic

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Q: If in a finite group every element is conjugate to its inverse, then show that all the entries in…

A: Let G be a finite group and g∈G.We have to show that if g and g-1 are conjugate then χ(g) is real…

Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: What are all abelian groups of order 81 up to isomorphism? Are there any Theorems I can use to…

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Q: Prove that if H is a subgroup of a finite group G of index p, where p is the smallest prime that…

A: This question is about group theory.

Q: 26. Show that any finite subgroup of the multiplicative group of a field is cyclic.

A: