Classification of groups of order 8

In summary, the conversation discusses a problem involving a group of order 8 and a specific element with order 4. The goal is to show that a subgroup of this group is normal and to list all possibilities for certain elements. The conversation also mentions examining pairs to show that the group is isomorphic to certain other groups. Finally, the group of order 8 is shown to be isomorphic to several other groups.
  • #1
mathusers
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Hi next one, bit confused with this problem: any hints on any of the parts would be greatly appreciated.

QUESTION:
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let G be a group of order 8 and suppose that [itex]y \epsilon G[/itex] has ord(y)=4. Put [itex]H = [1,y,y^2,y^3][/itex] and let [itex]x \epsilon G-H[/itex]

(i) show that [itex]H \lhd G[/itex] and that [itex]x^2 \epsilon H[/itex]
(ii) list, a priori, the possibilities for [itex]x^2[/itex] and label them [itex]P_1,...,P_n[/itex].
(iii) list the possibilities for [itex]xyx^{-1}[/itex] (HINT: CHECK ORDERS) and label them [itex]Q_1,...,Q_m[/itex]
(iv) By examining the pairs, [itex](P_i,Q_j)[/itex] in turn show that G is isomorphic to one of [itex]C_8, C_4 \times C_2, D_8, Q_8[/itex].
Deduce that an arbitrary group of order 8 is isomorphic to one of [itex]C_2 \times C_2 \times C_2, C_4 \times C_2, C_8, D_8, Q_8[/itex]
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For part (i) i understand i have to show G is a normal subgroup of H but I am not sure how to show that G is a subgroup (i.e. closure holds, existence of identity element of G in H, etc) and it is normal. i..e For all g in G, gHg^−1 ⊆ N.
 
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  • #2
mathusers said:
For part (i) i understand i have to show G is a normal subgroup of H but I am not sure how to show that G is a subgroup (i.e. closure holds, existence of identity element of G in H, etc) and it is normal. i..e For all g in G, gHg^−1 ⊆ N.
You want to show H is a normal subgroup of G not the reverse. Showing that H is a subgroup follows from the fact that y has order 4. Do you see why? To show it's a normal subgroup of G consider the left (or right) cosets of H in G. How many are there?
 

Related to Classification of groups of order 8

1. What is the definition of a group of order 8?

A group of order 8 refers to a mathematical structure consisting of 8 elements that follow a set of rules and operations, such as multiplication and addition, to form a group. In other words, it is a set of 8 objects that can be combined in a specific way to produce another object within the group.

2. How many groups of order 8 are there?

There are 5 distinct groups of order 8, known as the cyclic group, dihedral group, quaternion group, general linear group, and modular group. Each group has its own unique properties and characteristics.

3. What is the difference between a cyclic group and a dihedral group of order 8?

A cyclic group of order 8 is a group in which all elements can be generated by a single element through repeated multiplication. On the other hand, a dihedral group of order 8 is a group of symmetries of a regular octagon, where the elements represent rotations and reflections of the shape.

4. Can a group of order 8 have subgroups?

Yes, a group of order 8 can have subgroups. In fact, all groups of order 8 have at least 4 subgroups, including the trivial subgroup and the entire group itself. The number of subgroups a group of order 8 has depends on its structure and properties.

5. What are some real-world applications of groups of order 8?

Groups of order 8 have many applications in various fields, such as chemistry, physics, and computer science. For example, the dihedral group of order 8 can be used to model the symmetries of molecules, while the modular group is used in cryptography and coding theory.

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