Question

The subgroup relation ≤ on the set of subgroups G is reflexive, transitive, and anti-symmetric.

The subgroup relation ≤ on the set of subgroups G is reflexive, transitive, and anti-symmetric.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

Let H be a subgroup of G. Now, every group is a subgroup of itself. Hence, . Hence the subgroup relation is reflexive..

Now, let H1, H2, H3 be three subgroups of G such that . Now, since H1 is a subgroup of H2, and H2 is a subgroup of H3, therefore, H1 must be a subgroup of H3. That is, . Therefore, the subgroup relation is transitive.

Now, let . That is, H1 must also be a subset of H2 and H2 must be a subset of H1. This is only possible if H1=H2. Hence, the subgroup relation is anti-symmetric.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
show that the relation "≈" is reflexive, symmetric, and transitive on the class of all sets.
show that the relation "≈" is reflexive, symmetric, and transitive on the class of all sets.
Disprove: The following relation R on set Q is either reflexive, symmetric, or transitive. Let t...
Disprove: The following relation R on set Q is either reflexive, symmetric, or transitive. Let t and z be elements of Q. then t R z if and only if t = (z+1) * n for some integer n.
Determine whether the relation R is reflexive, symmetric, antisymmetric, and/or transitive [4 Marks] 22 The relation...
Determine whether the relation R is reflexive, symmetric, antisymmetric, and/or transitive [4 Marks] 22 The relation R on Z where (?, ?) ∈ ? if ? = ? . The relation R on the set of all subsets of {1, 2, 3, 4} where SRT means S C T.
Consider the relation R= {(1,2),(2,2),(2,3),(3,1),(3,3)}. Is R transitive, not reflexive, symmetric or equivalence relation?
Consider the relation R= {(1,2),(2,2),(2,3),(3,1),(3,3)}. Is R transitive, not reflexive, symmetric or equivalence relation?
4. What is a group action? Which subgroups of a symmetric group are called transitive? What...
4. What is a group action? Which subgroups of a symmetric group are called transitive? What is an example of a transitive subgroup of S6 that is not S6 itself?
Prove that strong connectivity is reflexive, transitive and symmetric.
Prove that strong connectivity is reflexive, transitive and symmetric.
Show that reflexive sentences are independent from symmetric, and transitive sentences by constructing a structure that...
Show that reflexive sentences are independent from symmetric, and transitive sentences by constructing a structure that satisfy symmetric and transitive quality but not reflexive. Reflexive :∀xE(x, x) symmetric :∀xy(E(x, y) → E(y, x)) transitive: ∀xyz(E(x, y) ∧ E(y, z) → E(x, z))
Show that symmetric sentences are independent from reflexive, and transitive sentences by constructing a structure that...
Show that symmetric sentences are independent from reflexive, and transitive sentences by constructing a structure that satisfy transitive and reflexive quality but not symmetric. Reflexive :∀xE(x, x) symmetric :∀xy(E(x, y) → E(y, x)) transitive: ∀xyz(E(x, y) ∧ E(y, z) → E(x, z))
Prove or disprove: The relation "is-a-normal-subgroup-of" is a transitive relation.
Prove or disprove: The relation "is-a-normal-subgroup-of" is a transitive relation.
For each of the properties reflexive, symmetric, antisymmetric, and transitive, carry out the following. Assume that...
For each of the properties reflexive, symmetric, antisymmetric, and transitive, carry out the following. Assume that R and S are nonempty relations on a set A that both have the property. For each of Rc, R∪S, R∩S, and R−1, determine whether the new relation must also have that property; might have that property, but might not; or cannot have that property. A ny time you answer Statement i or Statement iii, outline a proof. Any time you answer Statement ii,...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT