List The Left Cosets Of The Subgroups In Each Of The Followi
List the left cosets of the subgroups in each of the following. Here …
List the left cosets of the subgroups in each of the following. Here \( \langle a \rangle \) denotes the subgroup generated by the element \( a \).
(a) \( \langle 8 \rangle \) in \( (\mathbb{Z}_{24}, +) \)
(b) \( \langle 3 \rangle \) in \( U(8) \)
(c) \( 3 \mathbb{Z} \) in \( \mathbb{Z} \) (where \( 3\mathbb{Z} = \{3k : k \in \mathbb{Z}\} \)).
(d) \( A_n \) in \( S_n \) (where \( A_n \) is the set of all even permutations on \( \{1, \dots, n\} \)).
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In group theory, cosets are fundamental constructs that partition the parent group into disjoint subsets related to a subgroup. Understanding how to list left cosets in various familiar subgroups provides insights into the structure and properties of these groups.
For part (a), consider the subgroup \( \langle 8 \rangle \) within the additive group \( \mathbb{Z}_{24} \). Since \( \mathbb{Z}_{24} \) is the integers modulo 24 under addition, the subgroup generated by 8, denoted \( \langle 8 \rangle \), comprises the multiples of 8 modulo 24: \(\{0, 8, 16\}\). The left cosets of \( \langle 8 \rangle \) are formed by adding every element of \( \mathbb{Z}_{24} \) to the subgroup. The cosets are:
\[
0 + \langle 8 \rangle = \{0, 8, 16\}
\]
\[
1 + \langle 8 \rangle = \{1, 9, 17\}
\]
\[
2 + \langle 8 \rangle = \{2, 10, 18\}
\]
\[
3 + \langle 8 \rangle = \{3, 11, 19\}
\]
\[
4 + \langle 8 \rangle = \{4, 12, 20\}
\]
\[
5 + \langle 8 \rangle = \{5, 13, 21\}
\]
\[
6 + \langle 8 \rangle = \{6, 14, 22\}
\]
\[
7 + \langle 8 \rangle = \{7, 15, 23\}
\]
These eight cosets partition the group \( \mathbb{Z}_{24} \) into equal-sized subsets, each representing a distinct coset of the subgroup.
For part (b), examine \( U(8) \), the group of units modulo 8, which contains elements invertible modulo 8. The group \( U(8) = \{1, 3, 5, 7\} \). Each element generates a subgroup:
- \( \langle 1 \rangle = \{1\} \),
- \( \langle 3 \rangle = \{1, 3\} \),
- \( \langle 5 \rangle = \{1, 5\} \),
- \( \langle 7 \rangle = \{1, 7\} \).
The left cosets of these subgroups are:
- For \( 1 \): \( 1 \times \langle 1 \rangle = \{1\} \),
- For \( 3 \): \( 3 \times \langle 3 \rangle = \{3, 1\} \) (which is \( \{1, 3\} \)),
- For \( 5 \): \( 5 \times \langle 5 \rangle = \{5, 1\} \),
- For \( 7 \): \( 7 \times \langle 7 \rangle = \{7, 1\} \).
Since \( U(8) \) has four elements, each forms a coset with its respective subgroup, and these cosets partition \( U(8) \).
In part (c), the subgroup \( 3\mathbb{Z} \) within \( \mathbb{Z} \) consists of all integers divisible by 3, i.e., \( \dots, -6, -3, 0, 3, 6, \dots \). The cosets of this subgroup are formed by adding fixed integers to the subgroup, leading to three distinct cosets modulo 3:
\[
0 + 3\mathbb{Z} = \{ \dots, -6, -3, 0, 3, 6, \dots \}
\]
\[
1 + 3\mathbb{Z} = \{ \dots, -5, -2, 1, 4, 7, \dots \}
\]
\[
2 + 3\mathbb{Z} = \{ \dots, -4, -1, 2, 5, 8, \dots \}
\]
These partitions demonstrate how integers split into residue classes modulo 3.
Finally, for the symmetric group \( S_n \), the subgroup \( A_n \) (alternating group) consists of all even permutations of \( n \) elements. The left cosets of \( A_n \) form two classes because \( A_n \) is a normal subgroup of index 2:
\[
A_n \quad \text{and} \quad (12)A_n
\]
where \( (12) \) is a transposition. The cosets partition \( S_n \) into even and odd permutations, with each permutation belonging uniquely to one of these cosets.
These examples illuminate the concept of cosets as building blocks of group structure, partitioning groups into equal, disjoint subsets associated to subgroups, thereby underpinning essential results such as Lagrange’s theorem, which relates subgroup indices to the order of the group.
References
- Gallian, J. A. (2016). Contemporary Abstract Algebra (9th ed.). Brooks Cole.
- Dummit, D. S., & Foote, R. M. (2004). Abstract Algebra (3rd ed.). Wiley.
- Herstein, I. N. (2006). Topics in Algebra. Wiley.
- Fraleigh, J. B. (2003). A First Course in Abstract Algebra (7th ed.). Pearson.
- Lang, S. (2002). Algebra (3rd ed.). Springer.
- Rotman, J. J. (2012). Advanced Modern Algebra. Springer.
- Stillwell, J. (2003). Classical Topology and Geometry. Springer.
- Knapp, A. W. (2002). Basic Algebra. Springer.
- Hungerford, T. W. (2003). Abstract Algebra: An Introduction. Dover Publications.
- Fulton, W., & Harris, J. (1991). Representation Theory: A First Course. Springer.