# General linear group:GL(2,Z4)

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## Definition

This group can be defined in the following equivalent ways:

1. It is the group $GL(2,\mathbb{Z}_4)$ or $GL(2,\mathbb{Z}/4\mathbb{Z})$, i.e., the general linear group of degree two over the ring of integers modulo $4$.
2. It is the group $GL(2,\mathbb{F}_2[t]/(t^2))$, i.e., the general linear group of degree two over the ring $\mathbb{F}_2[t]/(t^2)$.

Note that although the rings in question are different, the corresponding general linear groups are isomorphic. This behavior is specific to the cases $p = 2$ and $p = 3$. The isomorphism can be traced to the fact that the quotient map: $GL(2,\mathbb{Z}/4\mathbb{Z}) \to GL(2,\mathbb{Z}/2\mathbb{Z})$

has a section, i.e., the corresponding short exact sequence splits. See also isomorphic general linear groups not implies isomorphic rings.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 96 groups with same order As $GL(2,R)$, $R$ a discrete valuation ring of length $l = 2$ with residue field of size $q = 2$: $q^{4l - 3}(q - 1)(q^2 - 1) = 2^5(1)(3) = 96$
exponent of a group 12 groups with same order and exponent of a group | groups with same exponent of a group
nilpotency class -- not a nilpotent group
derived length 3 groups with same order and derived length | groups with same derived length

## GAP implementation

### Group ID

This finite group has order 96 and has ID 195 among the groups of order 96 in GAP's SmallGroup library. For context, there are 231 groups of order 96. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(96,195)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(96,195);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [96,195]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description Functions used
GL(2,ZmodnZ(4)) GL and ZmodnZ