Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

Q: 3. Prove that a subset H of a finite group G is a subgroup of G if and only if a. His nonempty, and…

A: We have to prove given property:

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

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Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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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: Prove that a group of order 175 is Abelian.

A: Let G be a group of order 175 We first try to rewrite 175 as prime factorization as follows: 175 =…

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

A: Consider 2 ℤ = {an / n∈ ℤ }This is a subgroup of ℤ .claim : 2ℤ is isomorphic to ℤ.…

Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

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

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Q: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

A:

Q: If Φ is a homomorphism from Z30 onto a group of order 5, determinethe kernel of Φ.

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Q: Prove or Disprove that the Klein 4-group Va is isomorphic to Z4.

A: The statement is wrong.

Q: Let G be a group. Show that for all a, b E G, (ab)2 = a2b2 G is abelian

A: To prove that the group G is commutative (abelian) under the given conditions

Q: Show that if H is any group and h is an element of H, with h" = 1, then there is a unique…

A: Given that H is a group and h ∈H  Now,we define a mapping f:Z→H  such that   f(n) = hn for n∈Z For…

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: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

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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: 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: Prove that a group of order 15 is cyclic

A:

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: The center of an abelian group G is: a) {e) b) G c) A cyclic subgroup d) None of these

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Q: Prove that in a group, (a-1)-1 = a for all a.

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: Assume that G is a group such that for all x E G, * x = e. Prove that G is an abelian group.

A: Here we have to prove that G is an abelian group.

Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.

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Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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Q: If G is a cyclic group of order n, then G is isomorphic to Zn. true or false?

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Q: Let H be the set of all elements of the abelian group G that have finite order. Prove that H is a…

A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁1(G), then H≤ Z(G). n Core

A: Given that,  G is group of order 231 and H∈syl11G. We first claim that there is a unique Sylow…

Q: Let G be a group with center Z(G). Assume that the factor group G/Z(G) is cyclic. Prove that G is…

A: To prove that the group G is abelian if the quotient group G/Z(G) is cyclic, where Z(G) is the…

Q: Suppose H, and H2 subgroups of the group G. Prove hat H1 N Hzis a sub-group of G. are

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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: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

Q: not

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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…

A:

Q: Prove that if every non-identity element of a group G is of order 2,then G is abelion

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Q: prove that a group G of order p^2, where p is a prime, is abelian.

A: Suppose, G is a group of order p2 where p is prime.

Q: This is abstract algebra:   Prove that if "a" is the only elemnt of order 2 in a group, then "a"…

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Q: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

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Q: Let M be a group (not necesarily an Abelian group) of order 387. Prove that M must have an element…

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Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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Q: Let G be a group. V a, b, c d and x in G, if axb = cxd then ab = cd then G is necessarily:…

A: The answer is given as follows :

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…